#6366556

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

1597 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)(5,8)(9,12)(10,14)(11,13)(15,16)(18,24)(19,26)(20,28)(21,22)(25,29)(27,32);(1,3)(2,6)(4,11)(5,10)(7,13)(8,14)(9,15)(12,16)(17,18,23,19)(20,29,28,32)(21,26,22,24)(25,30,27,31);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,19,23,18)(20,27,28,25)(21,26,22,24)(29,31,32,30);(2,8)(6,14)(7,9)(13,15)(17,20)(18,25)(19,27)(21,30)(22,31)(23,28)(24,32)(26,29);(3,10)(6,14)(11,16)(13,15)(17,21,23,22)(18,26,19,24)(20,30,28,31)(25,29,27,32)

Polycyclic presentation:

2^9|b^2=h,c^2=h,e^2=h,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, 6273752);