#503141

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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 ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1184 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)(6,8)(9,10)(11,23)(12,27)(13,19)(14,33)(15,34)(16,36)(17,24)(18,30)(20,29)(21,28)(22,32)(25,40)(26,39)(31,38)(35,37);(1,2,4,3)(9,11,13,24)(10,17,19,23)(12,28,25,29)(14,31,15,35)(16,30,32,26)(18,36,39,22)(20,40,21,27)(33,37,34,38);(5,6,7,8)(9,12,29,11,16,31,15,26,13,25,28,24,32,35,14,30)(10,18,33,37,22,17,21,40,19,39,34,38,36,23,20,27);(1,3,4,2)(10,19)(17,23)(18,39)(20,21)(22,36)(27,40)(33,34)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410337);