#126472

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

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

391 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 56 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=i,c^2=i,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, 33668);