#6351164

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351101
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

2676 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258360);