#6351947

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351885
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2736 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:

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

Polycyclic presentation:

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

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, 6259143);