#6351806

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351787
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	#62755	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2689 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259002);