#503139

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×#3287	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

1192 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1192 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)(11,16)(12,24)(13,29)(14,32)(15,20)(17,26)(18,30)(19,21)(22,37)(23,35)(25,33)(27,34)(28,40)(31,36)(38,39);(1,2,4,3)(9,11,15,23)(10,16,20,35)(12,25,28,30)(13,27,31,38)(14,26,21,22)(17,19,37,32)(18,24,33,40)(29,34,36,39);(5,6,7,8)(9,12,25,23,14,27,13,26,15,28,30,11,21,38,31,22)(10,17,36,34,19,35,18,24,20,37,29,39,32,16,33,40);(1,3,4,2)

Polycyclic presentation:

2^9|b^2=ei,c^2=fhi,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, 410335);