#7143628

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

2149 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)(9,10,16,19)(11,15,20,14)(12,23,26,21)(13,18,29,35)(17,31,33,25)(22,28,36,30)(24,38,27,37)(32,39,34,40);(2,5)(3,7)(9,11,14,10,16,20,15,19)(12,23,25,33,26,21,31,17)(13,18,28,22,29,35,30,36)(24,34,39,38,27,32,40,37);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,21)(13,27)(14,25)(15,31)(16,26)(18,34)(19,33)(20,23)(22,38)(24,29)(28,40)(30,39)(32,35)(36,37);(2,5)(6,8)(9,13)(10,18)(11,22)(12,24)(14,30)(15,28)(16,29)(17,32)(19,35)(20,36)(21,37)(23,38)(25,40)(26,27)(31,39)(33,34);(1,3,4,7)(2,6,5,8)(10,19)(11,20)(17,33)(18,35)(21,23)(22,36)(32,34)(37,38)

Polycyclic presentation:

2^9|a^2=i,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, 7050824);