#503134

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
prime radical	C₂×#3283	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1166 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1166 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,d^2=e,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, 410330);