#6367127

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366870
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

1598 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:

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

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=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274323);