#6356827

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356733
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1584 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)(3,5)(4,12)(6,11)(7,9)(8,13)(10,15)(14,16)(17,18)(19,25)(20,26)(21,23)(22,30)(24,27)(28,32)(29,31);(1,3,5,2,7,11,6,9)(4,10,13,8,15,12,14,16)(18,23)(24,31)(25,32)(26,30);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,16)(11,12)(17,19)(18,24)(20,27)(21,28)(22,29)(23,31)(25,30)(26,32);(1,5,7,6)(2,3,9,11)(4,13,15,14)(8,10,16,12)(17,19)(18,25)(20,29)(21,28)(22,27)(23,32)(24,30)(26,31);(17,20,21,22)(18,26,23,30)(19,27,28,29)(24,32,31,25)

Polycyclic presentation:

2^9|b^2=fi,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dh,g^c=gi,e^d=ehi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264023);