#93340

order	512
nilpotency class	3
isoclinism	isoclinic class of #7035
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#95
commutator	C₂²×C₄	C₄×C₈
Frattini, cosocle	C₂²×C₄×C₈	C₂²
socle	C₂³	#336
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

270 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efgi,c^b=ci,d^b=dg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 536);