#93331

order	512
nilpotency class	4
isoclinism	isoclinic class of #7221
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#340
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₂²×C₄×C₈	C₂²
prime radical	C₂⁴	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 286 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fgi,d^2=h,e^2=fg,f^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fi,c^b=ci,d^b=dg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 527);