Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6355044 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1526 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,31)(25,35)(30,44)(32,39)(33,40)(36,41)(37,42)(38,46)(43,45)(47,48);(1,3,5,12,6,11,4,13)(2,7,8,10,9,15,14,16)(17,19,21,31,23,32,22,33)(18,24,27,29,28,40,26,39)(20,30,35,45,36,46,37,47)(25,38,34,43,42,44,41,48);(1,2)(3,7)(4,14)(5,8)(6,9)(10,12)(11,15)(13,16)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,41)(27,34)(28,42)(29,43)(31,45)(32,46)(33,47)(39,48)(40,44);(3,11)(7,15)(10,16)(12,13)(17,21,23,22)(18,26,28,27)(19,31,32,33)(20,35,36,37)(24,39,40,29)(25,41,42,34)(30,45,46,47)(38,48,44,43);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,16,15,10)(19,32)(20,36)(24,40)(25,42)(29,39)(31,33)(34,41)(35,37)Polycyclic presentation:
2^9|b^2=f,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,e^b=eh,e^c=eh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6262297);