#503246

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
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

1162 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)(11,21)(12,25)(13,29)(14,32)(15,20)(16,23)(17,27)(18,26)(19,30)(22,34)(24,38)(28,37)(31,33)(35,39)(36,40);(5,6)(7,8)(9,11,22,39,14,12,13,23,15,24,26,27,30,28,31,40)(10,16,33,25,19,35,18,21,20,36,29,37,32,17,34,38);(1,2,3,4)(6,8);(9,12,26,40,14,24,31,39,15,28,22,23,30,11,13,27)(10,17,29,21,19,16,34,37,20,35,33,38,32,36,18,25)

Polycyclic presentation:

2^9|b^2=ehi,c^2=f,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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410442);