#125677

order	512
nilpotency class	4
isoclinism	isoclinic class of #125644
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂×#97	C₂³
prime radical	#3150	C₂²
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 349 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=f,e^2=hi,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, 32873);