#500855

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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 ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

932 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,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, 408051);