#93403

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	C₄²×C₈	C₂²
socle, prime radical	C₂³	#336
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

232 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 512

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

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

Polycyclic presentation:

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

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, 599);