Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9416 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1004 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)(5,12,13,16)(6,9,14,15)(7,10)(8,11)(17,18)(19,20)(21,30)(22,34,37,44)(23,38)(24,28)(25,29)(26,32)(27,36,43,35)(31,46,48,40)(33,41)(39,42)(45,47,50,49);(3,5,7,15,8,13,17,9)(4,6,10,16,11,14,18,12)(19,21,23,32,25,33,24,39)(20,26,28,30,29,42,38,41)(22,31,36,49,37,48,35,47)(27,40,44,45,43,46,34,50);(3,6)(4,9)(5,10)(7,16)(8,14)(11,15)(12,17)(13,18)(19,22,25,37)(20,27,29,43)(21,31,33,48)(23,36,24,35)(26,40,42,46)(28,44,38,34)(30,45,41,50)(32,49,39,47)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107890);