#6357032

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356901
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

1547 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)(3,11)(4,9)(5,10)(6,14)(7,8)(12,16)(13,15)(17,18)(19,24)(20,22)(21,23)(25,28)(26,29)(27,30)(31,32);(1,3,5,13,7,12,6,9)(2,4,14,16,8,15,10,11)(18,22)(23,25)(24,29)(30,31);(2,8)(3,12)(9,13)(10,14)(18,23)(21,28)(22,25)(24,30)(27,32)(29,31);(1,4,7,15)(2,9,8,13)(3,10,12,14)(5,16,6,11)(17,19)(18,24)(20,26)(21,27)(22,29)(23,30)(25,31)(28,32);(1,5,7,6)(2,10,8,14)(3,13,12,9)(4,11,15,16)(19,26)(24,29)(27,32)(30,31)

Polycyclic presentation:

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

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, 6264228);