#126476

order	512
nilpotency class	4
isoclinism	isoclinic class of #126283
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1613
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

283 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 54 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)(3,7)(6,8)(9,10,13,28,14,18,29,19)(11,15,21,38,22,32,39,33)(12,23,27,17,26,16,25,34)(20,35,37,31,24,30,36,40);(1,2,4,6)(3,8,7,5)(9,11,14,22)(10,15,18,32)(12,24,26,20)(13,21,29,39)(16,30,23,35)(17,31,34,40)(19,33,28,38)(25,37,27,36);(3,7)(5,8)(10,16)(12,25)(13,29)(15,30)(17,28)(18,23)(19,34)(20,36)(21,39)(24,37)(26,27)(31,38)(32,35)(33,40)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fi,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=df,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, 33672);