#124825

order	512
nilpotency class	4
isoclinism	isoclinic class of #124679
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
socle	C₂²	#1467
prime radical	C₂³	#407
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 277 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,5)(4,7)(6,8)(9,10,13,19,14,18,29,35,15,20,27,37,12,16,26,31)(11,21,23,38,24,34,39,36,25,30,28,40,22,17,32,33);(2,5)(3,8)(6,7)(10,16,20,18)(13,27)(17,30,34,21)(19,31,37,35)(23,28)(26,29)(32,39)(33,40,36,38);(9,11,14,24,15,25,12,22)(10,17,16,30,20,34,18,21)(13,28,29,32,27,23,26,39)(19,36,31,38,37,33,35,40)

Polycyclic presentation:

2^9|a^2=e,b^2=fg,c^2=fgh,d^2=gi,e^2=h,h^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,d^b=dg,e^b=efgi,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32021);