#98839

order	512
nilpotency class	3
isoclinism	isoclinic class of #8626
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

434 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 126 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,c^2=h,d^2=g,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,e^c=eg,f^c=fgh

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, 6035);