#7182010

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

1693 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=hi,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,f^b=fi,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, 7089206);