#126507

order	512
nilpotency class	4
isoclinism	isoclinic class of #126351
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1616
commutator, Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

251 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 251 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgi,c^2=f,d^2=f,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gi,h^a=hi,d^b=dfi,e^b=egi,d^c=dg,e^c=ehi,h^c=hi,e^d=ei

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33703);