#6351833

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

2769 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)(4,9)(5,8)(6,10)(7,13)(12,16)(14,15)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,30)(29,44)(31,39)(32,38)(33,40)(34,41)(35,42)(37,45)(43,46)(47,48);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,11)(10,16)(17,19,23,32)(18,24,27,39)(20,29,35,46)(21,30,22,31)(25,37,42,48)(26,38,36,28)(33,45,34,47)(40,43,41,44);(1,3)(2,7)(4,13)(5,12)(6,14)(8,11)(9,15)(10,16)(17,20)(18,25)(19,29)(21,34)(22,33)(23,35)(24,37)(26,41)(27,42)(28,43)(30,47)(31,45)(32,46)(36,40)(38,44)(39,48);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,35,34)(24,38,39,28)(25,40,42,41)(29,45,46,47)(37,43,48,44);(1,4,6,5)(2,9,10,8)(3,13,14,12)(7,15,16,11)(17,21,23,22)(18,26,27,36)(19,31,32,30)(20,34,35,33)(24,28,39,38)(25,41,42,40)(29,45,46,47)(37,43,48,44)

Polycyclic presentation:

2^9|b^2=i,d^2=h,e^2=h,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, 6259029);