#503263

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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 ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1128 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=fi,d^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=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, 410459);