#6350835

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350717
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

2688 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)(7,13)(8,12)(10,15)(14,16)(17,18)(19,25)(20,30)(21,26)(22,32)(23,27)(24,28)(29,31);(4,11)(6,12)(10,15)(13,16)(17,19,23,29)(18,24,27,30)(20,21,28,22)(25,26,31,32);(1,2,5,8)(3,7,9,14)(4,12,11,6)(10,16,15,13)(17,20)(18,25)(19,21)(22,29)(23,28)(24,26)(27,31)(30,32);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,15)(12,16)(17,18)(19,24)(20,31)(21,32)(22,26)(23,27)(25,28)(29,30);(3,9)(7,14)(10,15)(13,16)(17,21,23,22)(18,26,27,32)(19,28,29,20)(24,31,30,25)

Polycyclic presentation:

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

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, 6258031);