#503100

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
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
prime radical	C₂×#3147	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1312 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)(9,10,15,19)(11,16,14,29)(12,22,26,37)(13,27,28,40)(17,31,32,24)(18,21,33,23)(20,34,25,39)(30,38,36,35);(1,2,4,3)(5,6)(7,8)(9,11,15,14)(10,16,19,29)(12,20,26,25)(13,21,28,23)(17,30,32,36)(18,27,33,40)(22,34,37,39)(24,35,31,38);(6,8)(9,12,23,38,14,25,28,31,15,26,21,35,11,20,13,24)(10,17,27,34,16,30,33,37,19,32,40,39,29,36,18,22);(1,3,4,2)

Polycyclic presentation:

2^9|a^2=i,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

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, 410296);