#6351817

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351817
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2897 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)(4,11)(7,15)(8,12)(10,13)(14,16)(17,18)(19,25)(20,28)(21,23)(22,27)(24,29)(26,32)(30,31);(1,2)(3,7)(4,12)(5,8)(6,11)(9,16)(10,14)(13,15)(20,29)(24,28)(25,32)(27,30);(1,3)(2,7)(4,13)(5,9)(6,14)(8,16)(10,11)(12,15)(17,19)(18,22)(20,30)(21,26)(23,31)(24,32)(25,28)(27,29);(3,9)(7,16)(10,13)(14,15)(18,23)(22,31)(24,28)(25,32);(2,8)(6,12)(7,16)(14,15)

Polycyclic presentation:

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

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, 6259013);