#72342095

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#24260
commutator	C₂×#41	C₂³×C₄
Frattini, cosocle	C₂²×#41	C₂⁴
socle	C₂	#360155
prime radical	C₂×#3025	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

3546 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:

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

Polycyclic presentation:

2^10|b^2=e,c^2=f,e^2=j,f^2=j,c^b=cf,d^a=dh,d^b=dg,f^b=fj,g^a=gi,h^b=hij,h^g=hj,i^d=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72342100

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