#512320

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2215	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 620 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)(14,20)(15,26)(16,21)(17,31)(18,30)(22,37)(23,24)(25,35)(27,36)(28,41)(29,40)(32,38)(33,34)(42,44);(5,7,8,11)(6,9,10,12)(13,14,21,34,18,24,31,35,19,25,17,23,30,33,16,20)(15,22,36,44,29,39,40,42,27,37,26,38,41,43,28,32);(6,10)(9,12)(15,27)(22,37)(26,36)(28,40)(29,41)(32,42)(38,44)(39,43);(2,4)(13,15,16,28,30,41,17,26,19,27,31,40,18,29,21,36)(14,22,20,32,33,43,23,38,25,37,35,42,24,39,34,44)

Polycyclic presentation:

2^9|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, 419516);