#512321

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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 ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 578 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) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=efh,d^2=e,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=ghi,h^a=hi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419517);