#6351128

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

2858 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:

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

Polycyclic presentation:

2^9|a^2=h,b^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, 6258324);