#6359243

order	512
nilpotency class	3
isoclinism	isoclinic class of #6359226
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

1702 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)(3,4)(5,13)(6,9)(7,12)(8,11)(10,16)(14,15)(17,18)(19,22)(20,23)(21,25)(24,28)(26,29)(27,31)(30,32);(1,3)(2,7)(4,13)(5,15)(6,11)(8,10)(9,14)(12,16)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30);(1,3)(2,8)(4,9)(5,12)(6,11)(7,10)(13,14)(15,16)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(1,4,6,8)(2,3,9,11)(5,7,16,14)(10,15,13,12);(2,9)(3,11)(10,13)(12,15)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6266439);