#126229

order	512
nilpotency class	4
isoclinism	isoclinic class of #125701
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×#95	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×#95 ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂×#95 ◁ G
LECS	G ▷ C₂×#95 ▷ C₂⁴ ▷ C₂ ▷ C₁

437 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 437 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=gi,d^2=f,e^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gi,d^b=dfi,e^b=egi,d^c=dg,e^c=ehi,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33425);