#512465

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 672 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) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=gi,c^2=fhi,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,d^b=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, 419661);