#72320614

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#495862
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	C₂×C₄×C₁₆	C₂⁴
socle	C₂	#72286742
prime radical	#512951	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#504 ◁ #479899 ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1286 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:

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

Polycyclic presentation:

2^11|a^2=h,b^2=ehk,c^2=f,d^2=gi,f^2=h,g^2=ij,i^2=jk,j^2=k,b^a=be,c^b=cij,d^b=dg,d^c=djk,e^a=ek,e^b=ek,f^b=fj,f^d=fk,g^b=gi,g^c=gk,h^b=hk,i^b=ij,j^b=jk

Conjugacy classes & character table

212 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321025, #72321722, #72322125, …
maximal normal subgroup of	#72321025, #72321722, #72322125, …

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, 1, F.8);;
gap> SetPower(coll, 2, F.5*F.8*F.11);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, 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.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.10*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);