#7143627

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993887
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2128 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)(11,23)(12,28)(13,19)(14,22)(15,21)(16,20)(17,26)(18,30)(24,34)(25,37)(27,29)(31,32)(33,40)(35,39)(36,38);(2,5)(3,7)(9,11,14,26,16,25,15,27)(10,17,21,23,20,29,22,37)(12,19,30,39,31,24,36,40)(13,32,33,18,34,28,35,38);(1,2)(3,6)(4,5)(7,8)(9,12)(10,18)(11,24)(13,17)(14,30)(15,36)(16,31)(19,25)(20,38)(21,28)(22,32)(23,33)(26,40)(27,39)(29,34)(35,37);(2,5)(6,8)(9,13)(10,19)(11,18)(12,29)(14,35)(15,33)(16,34)(17,31)(20,24)(21,40)(22,39)(23,30)(25,38)(26,32)(27,28)(36,37);(1,3,4,7)(2,6,5,8)(10,20)(11,25)(18,38)(19,24)(21,22)(26,27)(28,32)(39,40)

Polycyclic presentation:

2^9|b^2=f,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050823);