#7143617

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
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

2125 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)(11,21)(12,25)(13,18)(14,34)(15,19)(16,24)(17,29)(20,35)(22,28)(23,39)(26,33)(27,40)(30,36)(31,38)(32,37);(2,5)(3,7)(9,11,14,24,15,23,35,40)(10,16,20,21,19,27,34,39)(12,26,28,38,30,18,29,37)(13,25,31,17,33,36,32,22);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,18)(13,16)(14,28)(15,30)(19,22)(20,36)(21,31)(23,26)(24,37)(25,34)(27,33)(29,35)(32,39)(38,40);(2,5)(6,8)(9,13)(10,18)(11,22)(12,27)(14,32)(15,33)(16,30)(17,23)(19,26)(20,38)(21,28)(24,36)(25,40)(29,39)(31,35)(34,37);(1,3,4,7)(2,6,5,8)(10,19)(11,23)(17,22)(18,26)(20,34)(24,40)(25,36)(37,38)

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=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, 7050813);