#125222

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

508 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 108 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fgi,c^2=hi,d^2=gi,e^2=h,b^a=bd,d^a=df,f^a=fi,c^b=chi,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, 32418);