#7149421

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

1822 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)(9,10,15,21,16,20,14,32)(11,22,26,34,27,38,25,17)(12,28,29,35,13,19,31,18)(23,33,24,37,30,40,39,36);(2,5)(3,7)(9,11,15,26,16,27,14,25)(10,17,32,38,20,34,21,22)(12,23,31,39,13,30,29,24)(18,33,28,37,35,40,19,36);(1,2)(3,6)(4,5)(7,8)(9,12,16,13)(10,18,20,35)(11,23,27,30)(14,29,15,31)(17,33,34,40)(19,32,28,21)(22,36,38,37)(24,26,39,25);(2,5)(6,8)(9,13,16,12)(10,19,20,28)(11,24,27,39)(14,29,15,31)(17,33,34,40)(18,32,35,21)(22,37,38,36)(23,26,30,25);(1,3,4,7)(2,6,5,8)(10,20)(17,34)(18,35)(19,28)(21,32)(22,38)(33,40)(36,37)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=i,e^2=h,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, 7056617);