#509846

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

845 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 845 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=i,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417042);