#141859

order	512
nilpotency class	4
isoclinism	isoclinic class of #11651
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

362 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=defgi,c^2=e,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,f^b=fg,g^b=gi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49055);