#6349524

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349437
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#62815	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2823 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256720);