#6358865

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358842
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

1707 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)(3,6)(4,7)(5,9)(8,12)(10,13)(11,15)(14,16)(17,18,22,24)(19,26,27,29)(20,23,30,28)(21,31,32,25);(1,3,5,11)(2,6,9,15)(4,10,12,16)(7,13,8,14)(17,19,22,27)(18,20,24,30)(21,23,32,28)(25,29,31,26);(2,7)(4,12)(6,13)(8,9)(10,16)(14,15)(17,20,22,30)(18,19,24,27)(21,26,32,29)(23,25,28,31);(17,20,22,30)(18,23,24,28)(19,25,27,31)(21,29,32,26);(18,24)(19,27)(23,28)(25,31)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,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,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, 6266061);