#72306304

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#13083
commutator	C₂²×Q₈	C₂³×C₄
Frattini, cosocle	C₂×#124	C₂⁴
prime radical	C₂×#23999	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂×#117 ◁ C₂×#3510 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2779 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)(3,9)(4,19,21,42,22,37,23,58)(5,24,6,27)(7,13)(8,12)(10,20,38,28,40,57,39,25)(11,29,36,26)(14,44,46,62,48,59,47,64)(15,49,17,53)(16,34)(18,33)(30,43)(31,45,60,54,35,56,61,50)(32,55,41,51)(52,63);(1,3,8,16)(2,9,12,34)(4,14)(5,17,6,15)(7,18,30,52)(10,35)(11,41,36,32)(13,33,43,63)(19,44)(20,56)(21,46)(22,48)(23,47)(24,53,27,49)(25,54)(26,55,29,51)(28,50)(31,40)(37,59)(38,61)(39,60)(42,62)(45,57)(58,64);(1,4,5,25)(2,10,36,58)(3,14,15,50)(6,28,8,22)(7,21,29,20)(9,31,41,64)(11,42,12,40)(13,38,27,19)(16,48,17,54)(18,46,55,45)(23,26,57,30)(24,37,43,39)(32,62,34,35)(33,60,53,44)(47,51,56,52)(49,59,63,61);(1,5,8,6)(2,11,12,36)(3,15,16,17)(4,20,22,57)(7,29,30,26)(9,32,34,41)(10,37,40,19)(13,24,43,27)(14,45,48,56)(18,55,52,51)(21,28,23,25)(31,59,35,44)(33,49,63,53)(38,58,39,42)(46,54,47,50)(60,64,61,62)

Polycyclic presentation:

2^10|a^2=e,b^2=ij,c^2=egj,d^2=j,e^2=i,g^2=j,h^2=j,c^a=cg,c^b=cf,d^a=dhj,d^b=dj,d^c=dhj,e^c=eh,e^d=eij,f^a=fj,f^b=fj,f^c=fj,g^a=gh,g^b=gj,g^c=gh,g^e=gij,h^a=hi,h^c=hi,h^d=hj,h^g=hj,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306323, #72306356, #72306364, …
maximal normal subgroup of	#72306323, #72306356, #72306364, …

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