#6350451

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350397
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

2913 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^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=ei

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257647);