#6353324

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353301
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3066 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260520);