#93449

order	512
nilpotency class	4
isoclinism	isoclinic class of #1366
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#333
commutator	C₄²	C₄×C₈
Frattini, cosocle	C₄²×C₈	C₂²
socle	C₂	#7031
prime radical	C₂³	#344
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

204 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

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

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, 645);