#503273

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1060 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(2,4)(5,6)(7,20)(8,15)(9,31)(10,36)(11,18)(12,40)(13,19)(14,23)(16,22)(17,32)(21,53)(24,55)(25,57)(26,51)(27,44)(28,54)(29,49)(30,50)(33,42)(34,66)(35,47)(37,41)(38,56)(39,43)(45,58)(46,62)(48,63)(52,61)(59,68)(60,65)(64,67);(5,7,21,58,12,24,10,23,13,25,32,27,11,38,37,62)(6,14,41,55,18,45,17,20,19,46,36,56,40,44,53,57)(8,26,59,34,29,22,28,35,30,60,63,33,39,61,64,9)(15,47,67,16,43,66,48,51,50,31,54,52,49,42,68,65);(1,2,3,4)(5,8)(6,15)(7,22)(9,27)(10,28)(11,39)(12,29)(13,30)(14,42)(16,20)(17,48)(18,43)(19,50)(21,59)(23,33)(24,60)(25,61)(26,38)(31,44)(32,63)(34,62)(35,58)(36,54)(37,64)(40,49)(41,67)(45,47)(46,66)(51,56)(52,57)(53,68)(55,65);(5,9,32,60,12,34,37,61,13,35,21,26,11,33,10,22)(6,16,36,42,18,51,53,47,19,52,41,66,40,65,17,31)(7,8,27,63,24,29,62,64,25,30,58,59,38,39,23,28)(14,43,56,68,45,50,57,67,46,49,55,48,44,15,20,54)

Polycyclic presentation:

2^9|b^2=ehi,c^2=f,d^2=eh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410469);