#6350375

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

2806 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)(19,26)(20,32)(21,27)(22,28)(23,29)(24,31)(25,30);(1,2,5,8)(3,7,9,14)(4,12,11,6)(10,16,15,13)(17,19,23,20)(18,24,29,25)(21,30,22,31)(26,28,32,27);(17,19,23,20)(18,25,29,24)(21,30,22,31)(26,27,32,28);(1,3,5,9)(2,7,8,14)(4,10,11,15)(6,13,12,16)(17,20,23,19)(18,26,29,32)(21,30,22,31)(24,28,25,27);(3,9)(7,14)(10,15)(13,16)(17,21,23,22)(18,27,29,28)(19,30,20,31)(24,26,25,32)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=h,e^2=i,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, 6257571);