#125315

order	512
nilpotency class	4
isoclinism	isoclinic class of #125068
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#95	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂²×#95 ◁ G
LECS	G ▷ C₂×#95 ▷ C₂⁴ ▷ C₂ ▷ C₁

698 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

(2,6,7,8)(9,10,13,15,14,11,12,16)(17,18)(19,22,20,23)(21,24);(1,2,5,7)(3,6,4,8)(9,11,14,10)(12,15,13,16)(18,22)(19,20)(23,24);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,15,11,16)(17,19,21,20)(18,22,24,23)

Polycyclic presentation:

2^9|a^2=e,b^2=gi,c^2=g,d^2=g,e^2=h,b^a=bd,c^a=cf,d^a=df,f^a=fi,c^b=chi,d^b=dg,e^b=efg,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32511);