#503107

order	512
nilpotency class	4
isoclinism	isoclinic class of #501046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1319 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(2,3)(9,10)(11,16)(12,22)(13,27)(14,29)(15,19)(17,25)(18,23)(20,33)(21,32)(24,35)(26,39)(28,40)(30,34)(31,38)(36,37);(1,2,4,3)(5,6)(7,8)(9,11,15,14)(10,16,19,29)(12,20,24,26)(13,21,28,23)(17,30,31,36)(18,27,32,40)(22,33,35,39)(25,34,38,37);(6,8)(9,12,23,37,14,26,28,38,15,24,21,34,11,20,13,25)(10,17,27,33,16,30,32,35,19,31,40,39,29,36,18,22);(1,3,4,2)(12,24)(17,31)(20,26)(22,35)(25,38)(30,36)(33,39)(34,37)

Polycyclic presentation:

2^9|b^2=ei,c^2=fh,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^c=di

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, 410303);