#503257

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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	C₂²×#508	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1098 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1098 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410453);