#6356530

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356405
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1537 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

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, 6263726);