#506372

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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	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

648 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 162 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)(9,10,14,19)(11,21,23,39)(12,25,26,40)(13,29,31,34)(15,32,30,20)(16,33,35,24)(17,38,37,28)(18,22,36,27);(5,6,7,8)(9,11,22,38,15,12,13,24,14,23,27,28,30,26,31,33)(10,16,34,25,20,37,18,21,19,35,29,40,32,17,36,39);(2,4)(6,8)(11,23)(12,26)(16,35)(17,37)(21,39)(24,33)(25,40)(28,38);(9,12,27,33,15,23,31,38,14,26,22,24,30,11,13,28)(10,17,29,21,20,16,36,40,19,37,34,39,32,35,18,25)

Polycyclic presentation:

2^9|a^2=i,b^2=egh,d^2=eh,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, 413568);