#509606

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 662 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)(9,10,15,20)(11,21,18,36)(12,17,16,35)(13,28,31,24)(14,32,34,23)(19,27,38,30)(22,37,25,40)(26,39,29,33);(5,6)(7,8)(9,11,15,18)(10,13,20,31)(12,25,16,22)(14,33,34,39)(17,26,35,29)(19,37,38,40)(21,23,36,32)(24,27,28,30);(2,4)(6,8)(9,12,15,16)(10,17,20,35)(11,22,18,25)(13,29,31,26)(14,27,34,30)(19,32,38,23)(21,37,36,40)(24,39,28,33);(2,4)(9,13,30,37,16,29,34,36,15,31,27,40,12,26,14,21)(10,18,32,39,17,22,38,24,20,11,23,33,35,25,19,28)

Polycyclic presentation:

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

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, 416802);