#7003090

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1807 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)(3,5)(4,6)(7,8)(9,10,13,29,15,19,14,20)(11,21,24,16,26,37,25,32)(12,18,30,17,28,34,27,33)(22,31,39,36,38,40,23,35);(1,2)(3,7)(4,6)(5,8)(9,11,14,25,15,26,13,24)(10,16,29,37,19,32,20,21)(12,22,30,39,28,38,27,23)(17,31,18,35,33,40,34,36);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,22)(13,27)(14,30)(15,28)(16,31)(18,29)(19,33)(20,34)(21,36)(23,24)(25,39)(26,38)(32,40)(35,37);(3,8)(5,7)(9,12)(10,18)(11,23)(13,30)(14,27)(15,28)(16,31)(17,29)(19,34)(20,33)(21,35)(22,24)(25,38)(26,39)(32,40)(36,37);(2,6)(5,7)(10,19)(16,32)(17,33)(18,34)(20,29)(21,37)(31,40)(35,36)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6910286);