#6349872

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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2590 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=hi,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, 6257068);