Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#8490 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#538 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2030 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);(3,4,7,12,8,11,22,26)(5,13,16,27,15,9,20,24)(6,17,19,28,18,10,21,25)(14,29,31,34,30,23,32,33)(35,36)(37,45)(38,49,51,42)(39,52)(40,44)(41,47)(43,54)(46,60,62,59)(48,58)(50,57,66,64)(53,56)(55,61,63,65);(3,5,8,15)(4,9,11,13)(6,14,18,30)(7,20,22,16)(10,23,17,29)(12,27,26,24)(19,32,21,31)(25,34,28,33)(35,37,40,48)(36,41,44,56)(38,46,51,62)(39,53,54,47)(42,55,49,63)(43,58,52,45)(50,65,66,61)(57,60,64,59);(3,6)(4,10)(5,14)(7,21)(8,18)(9,23)(11,17)(12,28)(13,29)(15,30)(16,32)(19,22)(20,31)(24,34)(25,26)(27,33)(35,38)(36,42)(37,46)(39,50)(40,51)(41,55)(43,57)(44,49)(45,59)(47,61)(48,62)(52,64)(53,65)(54,66)(56,63)(58,60)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94539);