#6360737

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

1547 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=h,e^2=i,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, 6267933);