#7149410

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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1846 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,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, 7056606);