#93472

order	512
nilpotency class	4
isoclinism	isoclinic class of #7313
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 494 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)(9,10,13,17)(11,19,21,36)(12,23,24,34)(14,18,27,35)(15,28,29,38)(16,26,32,25)(20,30,37,33)(22,31,39,40);(9,11)(10,15)(12,20)(13,21,27,39)(14,22)(16,31)(17,33,35,29)(18,30)(19,23)(24,37,26,28)(25,38)(32,36,34,40)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=h,b^a=bc,c^a=cf,e^a=egi,f^a=fi,g^a=gi,c^b=cg,d^b=dfi,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, 668);