#6360923

order	512
nilpotency class	3
isoclinism	isoclinic class of #6360896
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1726 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=chi,d^b=df,f^b=fh,e^c=eh,g^c=gi,f^d=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6268119);