#6350362

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350061
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

2653 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257558);