Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349761 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62761 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2613 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,9)(3,10,12,15)(4,8,5,13)(7,14,16,11)(17,18,23,28)(19,29,33,39)(20,34,36,42)(21,27,22,26)(24,31,40,32)(25,37,41,35)(30,44,46,38)(43,45,48,47);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19,23,33)(18,24,28,40)(20,30,36,46)(21,31,22,32)(25,38,41,44)(26,39,27,29)(34,48,42,43)(35,45,37,47);(1,2)(3,7)(4,13)(5,8)(6,9)(10,11)(12,16)(14,15)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,34)(27,42)(28,41)(29,43)(31,45)(32,47)(33,46)(39,48)(40,44);(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,36,37)(24,39,40,29)(25,34,41,42)(30,45,46,47)(38,48,44,43);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(20,36)(25,41)(30,46)(34,42)(35,37)(38,44)(43,48)(45,47)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6257023);