Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33854 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#33855 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 66 points:
(1,2);(3,4,10,16)(5,17,22,50)(6,23,26,46)(7,27,32,44)(8,15,9,14)(11,35,39,20)(12,33,41,25)(13,30,45,31)(18,51,55,37)(19,56,59,36)(21,60,34,42)(24,62,57,40)(28,47,63,38)(29,43,64,66)(48,54,65,53)(49,52,61,58);(3,5,10,22)(4,11,16,39)(6,18,26,55)(7,28,32,63)(8,21,9,34)(12,36,41,56)(13,42,45,60)(14,38,15,47)(17,44,50,27)(19,57,59,24)(20,30,35,31)(23,61,46,49)(25,53,33,54)(29,52,64,58)(37,62,51,40)(43,48,66,65);(3,6)(4,12)(5,18)(7,24)(8,33)(9,25)(10,26)(11,36)(13,40)(14,46)(15,23)(16,41)(17,48)(19,28)(20,52)(21,54)(22,55)(27,43)(29,31)(30,64)(32,57)(34,53)(35,58)(37,42)(38,49)(39,56)(44,66)(45,62)(47,61)(50,65)(51,60)(59,63);(3,6)(4,12)(5,19)(7,29)(8,25)(9,33)(10,26)(11,37)(13,43)(14,23)(15,46)(16,41)(17,49)(18,28)(20,54)(21,52)(22,59)(24,31)(27,40)(30,57)(32,64)(34,58)(35,53)(36,42)(38,48)(39,51)(44,62)(45,66)(47,65)(50,61)(55,63)(56,60)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530850);