#99235

order	512
nilpotency class	3
isoclinism	isoclinic class of #99196
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

478 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 92 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)(3,5)(4,7)(6,8)(11,14,15,16)(17,18,21,27,22,25,20,26)(19,28,29,32,30,23,31,24);(9,10,13,12)(11,14,15,16)(17,19)(18,23)(20,29)(21,31)(22,30)(24,26)(25,28)(27,32);(2,5)(3,8)(6,7)(9,11,13,15)(10,14,12,16)(17,19)(18,24,25,32)(20,29)(21,31)(22,30)(23,26,28,27)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=dhi,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, 6431);