#7149418

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

1870 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)(9,10,15,20,16,21,14,32)(11,22,27,35,25,39,26,17)(12,28,30,36,13,19,31,18)(23,33,40,37,29,34,24,38);(2,5)(3,7)(9,11,15,27,16,25,14,26)(10,17,32,39,21,35,20,22)(12,29,31,40,13,23,30,24)(18,34,28,38,36,33,19,37);(1,2)(3,6)(4,5)(7,8)(9,12)(10,18)(11,23)(13,16)(14,30)(15,31)(17,33)(19,20)(21,36)(22,38)(24,27)(25,29)(26,40)(28,32)(34,35)(37,39);(2,5)(6,8)(9,13,16,12)(10,19,21,28)(11,24,25,40)(14,30,15,31)(17,34,35,33)(18,32,36,20)(22,38,39,37)(23,26,29,27);(1,3,4,7)(2,6,5,8)(11,25)(17,35)(22,39)(23,29)(24,40)(26,27)(33,34)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,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, 7056614);