Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
| direct product | C2×#33501 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C24 |
| commutator | C22 | C24×C8 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C23 | C23×C8 |
| prime radical | C24 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 50 points:
(1,2);(3,4)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20,22,27,23,28,32,40,24,26,33,38,34,39,44,48)(21,25,30,36,31,37,43,47,29,35,41,45,42,46,49,50);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21)(20,25)(22,30)(23,31)(24,29)(26,35)(27,36)(28,37)(32,43)(33,41)(34,42)(38,45)(39,46)(40,47)(44,49)(48,50);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(20,26)(21,29)(27,38)(28,39)(30,41)(31,42)(40,48)(43,49);(4,10)(8,11)(9,18)(16,17)(21,29)(25,35)(30,41)(31,42)(36,45)(37,46)(43,49)(47,50)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529896);