#512469

order	512
nilpotency class	4
isoclinism	isoclinic class of #511954
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

662 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 202 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)(7,10)(8,9)(11,12)(13,14,21,29)(15,24,34,39)(16,22,31,23)(17,27,18,37)(19,25,30,26)(20,38,36,28)(32,43,44,41)(33,42,35,40);(6,9)(10,12)(13,15,19,32,20,33,18,31,21,34,30,44,36,35,17,16)(14,22,27,40,28,41,26,39,29,23,37,42,38,43,25,24);(1,2,3,4)(13,15,30,44,20,33,17,16,21,34,19,32,36,35,18,31)(14,23,37,40,28,43,25,39,29,22,27,42,38,41,26,24);(5,7,8,11)(6,10,9,12)(13,16,17,35,36,44,30,34,21,31,18,33,20,32,19,15)(14,24,25,43,38,42,37,23,29,39,26,41,28,40,27,22)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fh,d^2=e,e^2=hi,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=ghi,h^a=hi,c^b=ci,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, 419665);