#126514

order	512
nilpotency class	4
isoclinism	isoclinic class of #126439
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

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

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

normal subgroups only

This group has 294 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fi,c^2=fg,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=eh,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, 33710);