#512532

order	512
nilpotency class	4
isoclinism	isoclinic class of #511948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

610 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 282 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) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

(2,4)(5,6,8,10)(7,11,12,9)(13,14)(15,23)(16,28)(17,20)(18,31)(19,25)(21,24)(22,29)(26,35)(27,42)(30,43)(32,41)(33,44)(34,39)(36,38)(37,40);(5,7,8,12)(6,9,10,11);(13,15,24,38,18,27,30,41,17,26,39,44,29,40,16,25)(14,19,28,37,22,33,34,35,20,32,43,42,31,36,21,23);(1,2,3,4)(14,20)(19,32)(21,34)(22,31)(23,35)(28,43)(33,36)(37,42)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fh,d^2=gi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,h^a=hi

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419728);