Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9369 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9369 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2);(3,4)(5,8,12,18)(6,11)(7,10)(9,13,17,14)(15,16)(19,20)(21,29)(22,33,35,26)(23,36)(24,28)(25,31)(27,38)(30,43,46,50)(32,44)(34,41,37,42)(39,45,49,47)(40,48);(3,5)(4,8,10,18)(6,14)(7,12)(9,16,17,11)(13,15)(19,21,23,31,24,32,38,48)(20,25,27,29,28,40,36,44)(22,30,37,47,35,46,34,45)(26,39,42,43,33,49,41,50);(3,5)(4,9)(6,14)(7,12)(8,16)(10,17)(11,18)(13,15)(19,22)(20,26)(21,30)(23,37)(24,35)(25,39)(27,42)(28,33)(29,43)(31,47)(32,46)(34,38)(36,41)(40,49)(44,50)(45,48)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107312);