#126463

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×D₄	SD₁₆
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 264 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=fh,d^2=f,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,h^a=hi,d^b=dfi,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=ehi,f^c=fi,g^c=gi,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, 33659);