#7149419

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

1848 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)(9,10,14,21,15,19,23,20)(11,22,25,16,26,35,40,34)(12,27,29,36,24,39,28,17)(13,18,31,37,32,33,30,38);(2,5)(3,7)(9,11,23,40,15,26,14,25)(10,16,21,35,19,34,20,22)(12,13,29,31,24,32,28,30)(17,33,39,37,36,18,27,38);(1,2)(3,6)(4,5)(7,8)(9,12,15,24)(10,17,19,36)(11,13,26,32)(14,28,23,29)(16,33,34,18)(20,27,21,39)(22,38,35,37)(25,30,40,31);(2,5)(6,8)(9,13)(10,18)(11,24)(12,26)(14,31)(15,32)(16,17)(19,33)(20,38)(21,37)(22,39)(23,30)(25,28)(27,35)(29,40)(34,36);(1,3,4,7)(2,6,5,8)(10,19)(16,34)(17,36)(18,33)(20,21)(22,35)(27,39)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^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, 7056615);