Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8516 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#8521 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2226 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)(5,11,13,8)(6,9)(7,10)(12,17,18,15)(14,16)(19,20)(21,29,32,41)(22,33,35,26)(23,27)(24,28)(25,31,39,36)(30,44)(34,48,50,40)(37,42)(38,45)(43,46)(47,49);(3,5)(4,8)(6,12)(7,13)(9,15)(10,11)(14,18)(16,17)(19,21,24,32)(20,25,28,39)(22,30,35,46)(23,36,37,31)(26,38,33,47)(27,41,42,29)(34,49,50,45)(40,44,48,43);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(19,22)(20,26)(21,30)(23,34)(24,35)(25,38)(27,40)(28,33)(29,43)(31,45)(32,46)(36,49)(37,50)(39,47)(41,44)(42,48)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94948);