#125238

order	512
nilpotency class	4
isoclinism	isoclinic class of #125068
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×#95 ▷ C₂⁴ ▷ C₂ ▷ C₁

496 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ghi,d^2=gi,e^2=h,b^a=bd,d^a=df,f^a=fi,c^b=ch,d^b=dg,e^b=efgi,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, 32434);