#126464

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

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

normal subgroups only

This group has 275 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)(9,10,14,20,15,21,32,19)(11,22,25,33,26,39,31,34)(12,18,28,17,29,38,13,30)(16,27,36,23,37,24,35,40);(1,2,5,7)(3,6,4,8)(9,11)(10,16)(12,27)(13,23)(14,31)(15,26)(17,33)(18,39)(19,36)(20,35)(21,37)(22,38)(24,29)(25,32)(28,40)(30,34);(1,3,5,4)(2,6,7,8)(10,17)(12,28)(13,29)(14,32)(16,33)(18,20)(19,38)(21,30)(22,36)(23,24)(25,31)(27,40)(34,37)(35,39)

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=fhi,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, 33660);