#72307106

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#8350
commutator	C₂×M₁₆	C₂³×C₄
Frattini, cosocle	C₂×#131	C₂⁴
socle	C₂	C₂×#8350
prime radical	#10577624	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

4003 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,14,7,31,28,57,8,16,29,39,9,15,30,45)(3,10,18,35,19,52,49,64,20,37,50,58,21,36,51,63)(4,13,23,11,26,40,55,41,5,12,25,43,24,44,27,42)(17,33,46,32,53,59,56,60,22,34,54,38,47,62,48,61);(1,3)(2,10)(4,22)(5,17)(6,18)(7,19)(8,20)(9,21)(11,38)(12,33)(13,34)(14,35)(15,36)(16,37)(23,54)(24,53)(25,46)(26,47)(27,56)(28,49)(29,50)(30,51)(31,52)(32,43)(39,58)(40,62)(41,61)(42,60)(44,59)(45,63)(48,55)(57,64);(1,4)(2,11,16,43)(3,17)(5,8)(6,27)(7,24)(9,26)(10,32,37,38)(12,14,13,39)(15,41,31,42)(18,48)(19,47)(20,22)(21,53)(23,30)(25,28)(29,55)(33,58,34,35)(36,60,52,61)(40,57,44,45)(46,51)(49,54)(50,56)(59,64,62,63);(2,12,16,13)(10,33,37,34)(11,14,43,39)(15,40,31,44)(32,58,38,35)(36,62,52,59)(41,57,42,45)(60,63,61,64)

Polycyclic presentation:

2^10|a^2=f,c^2=hj,d^2=hj,e^2=j,f^2=ij,g^2=j,i^2=j,c^a=cfgj,c^b=cj,d^a=de,e^a=ej,e^c=eh,f^c=fi,g^a=gj,g^d=ghj,g^e=gj,h^a=hj,i^c=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307167, #72307431
maximal normal subgroup of	#72307167, #72307431

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