#6360717

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

1575 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)(3,10)(4,8)(5,13)(6,9)(7,12)(11,16)(14,15)(17,18,23,26)(19,29,30,31)(20,25,24,32)(21,28,22,27);(3,11)(7,15)(10,16)(12,14)(17,19)(18,24)(20,26)(21,32)(22,25)(23,30)(27,31)(28,29);(1,3,6,11)(2,7,9,15)(4,12,5,14)(8,16,13,10)(17,20)(18,19)(21,31)(22,29)(23,24)(25,27)(26,30)(28,32);(17,20)(18,25)(19,27)(21,29)(22,31)(23,24)(26,32)(28,30);(1,4,6,5)(2,8,9,13)(3,12,11,14)(7,16,15,10)(18,26)(19,30)(25,32)(27,28)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,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, 6267913);