#93385

order	512
nilpotency class	4
isoclinism	isoclinic class of #1366
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#333
commutator	C₄²	C₄×C₈
Frattini, cosocle	C₂²×C₄×C₈	C₂²
socle	C₂²	#1296
prime radical	C₂⁴	#101
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

356 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 122 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)(4,7)(6,8)(9,10,12,18,14,17,15,29)(11,20,21,32,24,31,23,16)(13,26,25,33,27,34,28,19)(22,30,36,35,38,40,37,39);(2,5,7,6)(9,11,13,22,15,21,28,36,14,24,27,38,12,23,25,37)(10,16,19,30,18,31,26,39,17,32,33,40,29,20,34,35)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=i,e^2=h,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgi,g^a=gi,h^a=hi,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, 581);