#6349714

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349684
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2653 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(3,10)(4,5)(7,14)(8,12)(11,13)(15,16)(17,18,23,28)(19,29,33,38)(20,25,36,42)(21,27,22,26)(24,32,39,31)(30,43,46,47)(34,41,35,40)(37,44,48,45);(1,2)(3,7)(4,8)(5,12)(6,9)(10,15)(11,16)(13,14)(17,19,23,33)(18,24,28,39)(20,30,36,46)(21,31,22,32)(25,37,42,48)(26,29,27,38)(34,44,35,45)(40,47,41,43);(1,3)(2,7)(4,10)(5,13)(6,11)(8,15)(9,16)(12,14)(17,20)(18,25)(19,30)(21,35)(22,34)(23,36)(24,37)(26,41)(27,40)(28,42)(29,43)(31,45)(32,44)(33,46)(38,47)(39,48);(1,4,6,5)(2,8,9,12)(3,10,11,13)(7,15,16,14)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,34,36,35)(24,29,39,38)(25,40,42,41)(30,44,46,45)(37,47,48,43);(17,21,23,22)(18,26,28,27)(19,32,33,31)(20,35,36,34)(24,38,39,29)(25,41,42,40)(30,44,46,45)(37,47,48,43)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256910);