Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#11461 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
834 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)(19,20,22,26,23,27,32,38,24,28,33,39,34,40,44,48)(21,25,30,36,31,37,43,47,29,35,41,45,42,46,49,50);(3,5)(4,8)(6,12)(7,13)(9,15)(10,11)(14,18)(16,17)(19,21)(20,25)(22,30)(23,31)(24,29)(26,36)(27,37)(28,35)(32,43)(33,41)(34,42)(38,47)(39,45)(40,46)(44,49)(48,50);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(21,29)(25,35)(30,41)(31,42)(36,45)(37,46)(43,49)(47,50)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387252);