#6357524

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357516
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2064 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=fh,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,e^b=eh,e^c=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264720);