#500665

order	512
nilpotency class	4
isoclinism	isoclinic class of #499897
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1167 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)(9,10,14,20)(11,23,12,17)(13,22,16,19)(15,30,29,33)(18,25,37,27)(21,28,39,32)(24,34,26,38)(31,35,40,36);(1,2,4,3)(5,6)(7,8)(9,11,14,12)(10,17,20,23)(13,26,16,24)(15,25,29,27)(18,33,37,30)(19,38,22,34)(21,35,39,36)(28,31,32,40);(6,8)(9,12,14,11)(10,18,20,37)(13,26,16,24)(15,31,29,40)(17,33,23,30)(19,36,22,35)(21,34,39,38)(25,32,27,28);(1,3,4,2)(9,13,14,16)(10,19,20,22)(11,24,12,26)(15,28,29,32)(17,34,23,38)(18,36,37,35)(21,33,39,30)(25,40,27,31)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,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^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407861);