#151395

order	512
nilpotency class	5
isoclinism	isoclinic class of #151158
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2196
commutator, Frattini, cosocle	#384	C₂³
prime radical	C₂³	B(4,𝔽₂)
LCS	G ▷ #384 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ G
LECS	G ▷ #384 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

302 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gh,d^b=df,e^b=eghi,h^b=hi,d^c=dg,e^c=eh,g^c=gi,e^d=eh,g^d=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58591);