#232713

order	512
nilpotency class	3
isoclinism	isoclinic class of #232706
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^b=eh,g^b=gh,e^c=ei,f^c=fi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139909);