#6366693

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366542
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

1573 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,e^c=eh,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, 6273889);