#6349847

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349830
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	#60907	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2617 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)(3,10)(4,7)(5,9)(6,14)(8,13)(11,15)(12,16)(17,18)(19,28)(20,30)(21,27)(22,32)(23,26)(24,29)(25,31);(1,3)(2,8)(4,11)(5,13)(6,15)(7,12)(9,16)(10,14)(17,19,23,25)(18,24,26,20)(21,29,22,30)(27,31,32,28);(1,4,7,2)(3,11,12,8)(5,9,6,14)(10,13,16,15)(17,20,23,24)(18,25,26,19)(21,31,22,28)(27,29,32,30);(3,12)(8,11)(10,16)(13,15)(18,26)(19,25)(27,32)(29,30);(1,5,7,6)(2,9,4,14)(3,13,12,15)(8,16,11,10)(17,21,23,22)(18,27,26,32)(19,29,25,30)(20,28,24,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257043);