#6350963

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350909
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

2622 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)(3,9)(4,6)(5,13)(7,11)(8,15)(10,14)(12,16)(18,24)(19,26)(20,28)(21,22)(25,27)(29,32);(1,3,6,12)(2,7,4,10)(5,14,15,11)(8,16,13,9)(17,18)(19,21)(20,25)(22,24)(23,26)(27,30)(28,32)(29,31);(1,4,6,2)(3,10,12,7)(5,13,15,8)(9,11,16,14)(17,19,23,24)(18,21,26,22)(20,29,28,27)(25,31,32,30);(2,4)(7,10)(8,13)(9,16)(17,20)(18,25)(19,27)(21,30)(22,31)(23,28)(24,29)(26,32);(20,28)(25,32)(27,29)(30,31)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,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, 6258159);