Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#13503 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #2077 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
670 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);(3,4,10,32,29,13,8,14,9,16,34,42,7,15,31,44)(5,17,33,63,19,11,20,47,22,41,51,36,21,37,53,66)(6,23,30,40,25,59,26,39,27,56,60,12,28,43,61,45)(18,48,52,65,54,55,50,35,24,57,62,46,49,64,58,38);(3,5)(4,11)(6,24)(7,21)(8,20)(9,22)(10,33)(12,38)(13,41)(14,36)(15,17)(16,37)(18,27)(19,29)(23,55)(25,49)(26,58)(28,54)(30,62)(31,53)(32,47)(34,51)(35,40)(39,46)(42,66)(43,48)(44,63)(45,65)(50,61)(52,60)(56,64)(57,59);(3,6,7,25,9,27,29,28)(4,12,13,39,16,40,15,45)(5,18,19,49,22,24,21,54)(8,30,10,26,31,60,34,61)(11,35,17,46,37,38,41,65)(14,43,42,56,44,59,32,23)(20,52,51,58,53,62,33,50)(36,57,47,64,63,48,66,55)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406587);