#126490

order	512
nilpotency class	4
isoclinism	isoclinic class of #126439
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1613
commutator, Frattini, cosocle	C₂×#97	C₂³
prime radical	#31207	C₂
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 325 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=i,d^2=f,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,g^a=gi,d^b=df,e^b=egi,f^b=fi,d^c=dg,e^c=eh,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, 33686);