#6351815

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

2806 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)(3,11)(4,10)(5,8)(6,9)(7,12)(13,16)(14,15)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,36)(30,38)(31,40)(33,39)(35,41)(37,42)(43,45)(44,47)(46,48);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,15)(10,16)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,38)(26,29)(27,39)(28,40)(34,47)(35,44)(36,45)(37,46)(41,48)(42,43);(4,5)(8,10)(11,16)(12,14)(17,20)(18,25)(19,30)(21,36)(22,35)(23,37)(24,38)(26,42)(27,34)(28,41)(29,43)(31,45)(32,44)(33,46)(39,47)(40,48);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,11,15,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,29,40,39)(25,34,41,42)(30,44,46,45)(38,47,48,43);(2,9)(7,15)(8,10)(11,16)(17,21,23,22)(18,27,28,26)(19,31,33,32)(20,36,37,35)(24,39,40,29)(25,34,41,42)(30,45,46,44)(38,47,48,43)

Polycyclic presentation:

2^9|d^2=h,e^2=hi,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, 6259011);