#7003088

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

1786 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)(9,10,15,20,14,19,13,21)(11,22,30,33,26,34,25,16)(12,18,27,17,29,37,28,35)(23,32,24,36,39,31,38,40);(1,2)(3,7)(4,6)(5,8)(9,11,13,25,14,26,15,30)(10,16,20,22,19,33,21,34)(12,23,27,24,29,39,28,38)(17,31,18,36,35,32,37,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,23)(13,27)(14,29)(15,28)(16,31)(18,20)(19,35)(21,37)(22,36)(24,25)(26,39)(30,38)(32,33)(34,40);(3,8)(5,7)(9,12)(10,18)(11,24)(13,28)(14,29)(15,27)(16,32)(17,20)(19,37)(21,35)(22,36)(23,25)(26,38)(30,39)(31,33)(34,40);(2,6)(5,7)(10,19)(16,33)(17,35)(18,37)(20,21)(22,34)(31,32)(36,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,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, 6910284);