#93267

order	512
nilpotency class	4
isoclinism	isoclinic class of #360
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#102
commutator	C₈	C₈²
Frattini, cosocle	C₄²×C₈	C₂²
socle	C₂²	#1279
prime radical	C₂³	#339
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

256 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 118 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)(11,19,21,31)(12,23)(13,27)(14,18)(15,20,29,24)(16,25)(17,28)(22,37,36,34)(26,33)(30,38,40,35)(32,39);(9,11)(10,15,17,30,18,29,27,40)(12,24)(13,22)(14,21)(16,31,23,37,32,19,33,34)(20,26)(25,38)(28,36)(35,39)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,b^a=bc,c^a=cf,e^a=efi,f^a=fi,h^a=hi,c^b=ci,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 463);