#125270

order	512
nilpotency class	4
isoclinism	isoclinic class of #125269
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

259 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=f,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cf,d^a=df,f^a=fi,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=di

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, 32466);