#72321447

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#151085
commutator	C₃₂	C₂³×C₈
Frattini, cosocle	#1397	C₂⁴
prime radical	C₂×#13742	C₄
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #11399 ◁ #149496 ◁ G
LECS	G ▷ #1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

769 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 128 points:

(2,10)(3,17)(4,23)(5,9)(6,26)(8,35)(11,45)(12,14)(13,54)(15,61)(16,53)(18,68)(19,71)(20,74)(21,76)(22,65)(24,81)(25,34)(27,93)(28,79)(29,96)(30,77)(31,92)(32,38)(33,83)(36,91)(37,90)(39,66)(40,67)(41,48)(42,57)(43,56)(46,109)(47,107)(49,86)(50,113)(51,111)(52,84)(55,106)(58,120)(59,89)(62,104)(63,108)(64,70)(69,82)(72,126)(73,125)(75,110)(78,80)(85,123)(87,102)(88,103)(94,99)(95,128)(97,101)(98,127)(100,118)(105,117)(112,121)(114,124)(115,122)(116,119);(1,2,5,14,7,12,9,10)(3,18)(4,24,27,13,28,42,30,52)(6,33,31,43,36,15,37,59)(8,16,32,60,38,53,35,44)(11,46)(17,64,65,108)(19,62)(20,51)(21,70)(22,50)(23,54,77,81,79,84,93,57)(25,86,87,121)(26,89,90,61,91,56,92,83)(29,85,78,58,95,105,98,116)(34,100)(39,47)(40,63)(41,75)(45,74,107,110)(48,109,111,66)(49,99)(55,71,115,125)(67,68,76,113)(69,73)(72,122)(80,117,96,120,127,123,128,119)(82,97,104,126)(88,118,94,124)(101,106)(102,114)(103,112);(1,3)(2,11)(4,25)(5,21)(6,34)(7,22)(8,39)(9,40)(10,41)(12,47)(13,55)(14,51)(15,62)(16,63)(17,23)(18,60)(19,26)(20,35)(24,82)(27,94)(28,87)(29,97)(30,88)(31,99)(32,75)(33,69)(36,102)(37,103)(38,46)(42,104)(43,106)(44,50)(45,54)(48,57)(49,56)(52,115)(53,70)(58,121)(59,122)(61,100)(64,119)(65,79)(66,96)(67,93)(68,123)(71,78)(72,90)(73,91)(74,80)(76,77)(81,111)(83,114)(84,107)(85,124)(86,116)(89,112)(92,101)(95,126)(98,125)(105,118)(108,120)(109,128)(110,127)(113,117);(2,12)(10,14)(11,47)(13,52)(15,33)(16,53)(18,50)(24,42)(41,51)(43,59)(44,60)(45,107)(48,111)(49,112)(54,84)(55,115)(56,89)(57,81)(58,116)(61,83)(62,69)(63,70)(64,108)(68,113)(82,104)(85,105)(86,121)(100,114)(106,122)(117,123)(118,124)(119,120)

Polycyclic presentation:

2^11|b^2=f,e^2=gi,f^2=h,g^2=ij,i^2=jk,j^2=k,b^a=bj,c^a=ce,c^b=cik,d^b=dk,e^a=eg,e^b=ejk,e^c=eg,f^a=fk,f^c=fj,f^e=fk,g^a=gi,g^b=gk,g^c=gi,h^c=hk,i^a=ij,i^c=ij,j^a=jk,j^c=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321544, #72323081, #72323087
maximal normal subgroup of	#72321544, #72323081, #72323087

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.10*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);