#72321671

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72286690
commutator	C₂×C₃₂	C₂²×C₈
Frattini, cosocle	C₂×#1397	C₂³
prime radical	C₂×#13742	C₄
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #11399 ◁ #149496 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

609 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 128 points:

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

Polycyclic presentation:

2^11|b^2=f,d^2=jk,e^2=gi,f^2=h,g^2=ij,i^2=jk,j^2=k,b^a=bj,c^a=ce,c^b=cd,d^a=dj,d^b=dk,d^c=dj,e^a=eg,e^b=ejk,e^c=eg,f^a=fk,f^c=fjk,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

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321722, #72323055, #72323081
maximal normal subgroup of	#72321722, #72323055, #72323081

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, 4, F.10*F.11);;
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.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
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*F.11);;
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);