#93471

order	512
nilpotency class	4
isoclinism	isoclinic class of #7310
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂⁴	C₄×C₈
Frattini, cosocle	C₂²×#117	C₂²
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×#117 ▷ C₂⁴ ▷ C₂ ▷ C₁

460 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=h,e^2=i,b^a=bc,c^a=cf,e^a=eg,f^a=fi,g^a=gi,c^b=cg,d^b=df,f^b=fi,d^c=di,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 667);