#6360712

order	512
nilpotency class	3
isoclinism	isoclinic class of #6360376
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

1652 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)(3,8)(4,9)(5,7)(6,10)(11,16)(12,15)(13,14)(17,18)(19,26)(20,24)(21,29)(22,25)(23,28)(27,32)(30,31);(2,7)(3,11)(4,5)(8,15)(9,10)(12,13)(19,27)(23,31)(26,32)(28,30);(17,19,22,27)(18,23,25,31)(20,28,21,30)(24,32,29,26);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15);(1,4,6,5)(2,7,10,9)(3,11,12,13)(8,14,15,16)(17,20,22,21)(18,24,25,29)(19,28,27,30)(23,32,31,26)

Polycyclic presentation:

2^9|c^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=eh,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, 6267908);