Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11867 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#11869 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1519 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C2×M16 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
984 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)(5,16,19,47)(6,22)(7,28,31,45)(8,14)(9,34)(10,15)(11,18)(12,26)(13,32,43,33)(17,48,50,64)(20,41,21,55)(23,58,51,46)(24,42,30,39)(25,40)(27,57)(29,44)(35,59)(36,53)(37,54)(38,52)(49,65)(56,60,62,61)(63,66);(3,5,10,19)(4,11,15,37)(6,23,27,51)(7,29,31,53)(8,21,9,20)(12,38,40,65)(13,41,43,55)(14,44,34,36)(16,45,47,28)(17,26,50,25)(18,32,54,33)(22,35,57,63)(24,49,30,52)(39,46,42,58)(48,56,64,62)(59,60,66,61);(3,6,8,25,10,27,9,26)(4,12,34,57,15,40,14,22)(5,17,20,23,19,50,21,51)(7,30,32,61,31,24,33,60)(11,35,44,65,37,63,36,38)(13,42,45,56,43,39,28,62)(16,46,55,64,47,58,41,48)(18,52,53,66,54,49,29,59)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389226);