Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#13230 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1964 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 66 points:
(1,2);(3,4)(5,15,20,38)(6,21)(7,26,16,12)(8,29)(9,14)(10,18)(11,24)(13,22)(17,41,45,64)(19,47,42,37)(23,52,39,49)(25,50)(27,32)(28,35,40,56)(30,59)(31,57)(33,54,55,63)(34,51)(36,61,62,44)(43,48)(46,58)(53,65)(60,66);(3,5,16,18,9,20,7,27)(4,10,26,38,14,32,12,15)(6,17,39,43,25,45,23,53)(8,19,40,46,22,42,28,57)(11,30,55,62,34,60,33,36)(13,31,56,47,29,58,35,37)(21,48,52,64,50,65,49,41)(24,44,63,59,51,61,54,66);(3,6,22,51,9,25,8,24)(4,11,29,50,14,34,13,21)(5,17,42,61,20,45,19,44)(7,23,40,63,16,39,28,54)(10,30,58,65,32,60,31,48)(12,33,56,52,26,55,35,49)(15,36,47,64,38,62,37,41)(18,43,57,66,27,53,46,59)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402390);