#6358429

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358426
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

2023 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265625);