#6360723

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=h,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, 6267919);