#99231

order	512
nilpotency class	3
isoclinism	isoclinic class of #99196
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

736 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=dh,e^2=gi,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fg,e^c=ei,f^c=fghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6427);