#6350563

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350525
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

2934 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)(9,10,14,18)(11,19,21,32)(12,16,23,30)(13,25,26,17)(15,27,29,22)(20,33,34,40)(24,37,36,31)(28,38,39,35);(1,2,5,7)(3,6,4,8)(13,26)(17,25)(19,32)(22,27)(24,36)(31,37)(33,40)(35,38);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,20)(13,27)(14,21)(16,28)(17,32)(18,29)(19,25)(22,26)(23,34)(24,38)(30,39)(31,40)(33,37)(35,36);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,20)(13,24)(14,23)(15,28)(17,31)(18,30)(19,33)(21,34)(22,35)(25,37)(26,36)(27,38)(29,39)(32,40);(12,23)(16,30)(20,34)(24,36)(28,39)(31,37)(33,40)(35,38)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257759);