#6362635

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362510
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

1753 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6269831);