Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357427 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,7,10)(3,5,9,6)(4,8,12,15)(11,13,16,14)(17,18,23,28)(19,29,31,40)(20,34,35,42)(21,27,22,26)(24,33,39,32)(25,36,41,37)(30,38,44,48)(43,45,47,46);(1,3,6,2,7,9,5,10)(4,11,14,8,12,16,13,15)(17,19,21,32,23,31,22,33)(18,24,26,29,28,39,27,40)(20,30,36,45,35,44,37,46)(25,38,42,43,41,48,34,47);(1,4)(2,8)(3,11)(5,13)(6,14)(7,12)(9,16)(10,15)(17,20,22,37,23,35,21,36)(18,25,27,34,28,41,26,42)(19,30,33,46,31,44,32,45)(24,38,40,47,39,48,29,43);(4,12)(8,15)(11,16)(13,14)(19,31)(20,35)(24,39)(25,41)(29,40)(32,33)(34,42)(36,37);(1,5,7,6)(2,9,10,3)(4,13,12,14)(8,16,15,11)(19,31)(24,39)(29,40)(30,44)(32,33)(38,48)(43,47)(45,46)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^b=eh,d^c=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264652);