#72307107

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#24338
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂	C₂×#24338
prime radical	C₂×#3414	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2961 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=g,c^2=g,d^2=gj,e^2=f,f^2=i,h^2=ij,i^2=j,b^a=bj,d^a=dgj,d^c=dgj,e^a=ef,e^c=efhi,e^d=efgh,f^a=fi,f^c=fj,f^d=fj,g^e=gj,h^a=hij,h^c=hj,i^a=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307129, #72307810, C₂×#72307107
maximal normal subgroup of	#72307129, #72307810, C₂×#72307107
maximal quotient of	C₂×#72307107

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