#7489884

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1677 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7397080);