#7143621

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

2154 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=f,e^2=h,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, 7050817);