#6351863

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351817
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

2807 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,29)(20,33)(21,27)(22,28)(23,26)(24,31)(25,35)(30,38)(32,40)(34,41)(36,39)(37,42)(43,46)(44,48)(45,47);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,24)(20,30)(21,36)(22,31)(23,32)(25,38)(26,39)(27,29)(28,40)(33,47)(34,44)(35,46)(37,45)(41,43)(42,48);(4,5)(8,13)(10,15)(11,14)(17,20)(18,25)(19,30)(21,35)(22,37)(23,34)(24,38)(26,42)(27,41)(28,33)(29,43)(31,45)(32,44)(36,46)(39,48)(40,47);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,26)(20,34)(24,39)(27,28)(29,40)(30,44)(35,37)(45,46);(3,12)(7,16)(10,15)(11,14)(17,21,23,22)(18,27,26,28)(19,31,32,36)(20,35,34,37)(24,40,39,29)(25,41,42,33)(30,45,44,46)(38,47,48,43)

Polycyclic presentation:

2^9|d^2=i,e^2=hi,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,g^a=ghi,e^b=eh,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, 6259059);