#512326

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

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^b=di,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, 419522);