#6362459

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362318
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,f^b=fh,g^c=gi,f^d=fh

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, 6269655);