#6350670

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

3754 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,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, 6257866);