Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#12995 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #1874 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1352 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);(3,4,9,16,10,17,34,48,11,18,33,47,8,15,7,14)(5,19,23,41,24,45,21,38,25,52,31,12,22,39,59,40)(6,26,29,46,28,44,32,50,30,49,36,42,35,13,27,43)(20,53,56,64,55,63,54,65,57,66,61,62,60,51,58,37);(3,5,10,22,11,25,8,24)(4,12,17,38,18,41,15,40)(6,20,28,60,30,57,35,55)(7,31,9,21,34,23,33,59)(13,37,26,62,44,65,49,64)(14,45,16,19,48,39,47,52)(27,61,29,54,32,56,36,58)(42,66,43,63,46,53,50,51);(3,6)(4,13)(5,20)(7,32)(8,35)(9,36)(10,28)(11,30)(12,37)(14,46)(15,49)(16,50)(17,26)(18,44)(19,51)(21,58)(22,60)(23,61)(24,55)(25,57)(27,34)(29,33)(31,56)(38,62)(39,66)(40,64)(41,65)(42,48)(43,47)(45,53)(52,63)(54,59)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401646);