#6585837

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574086
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	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

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, 6493033);