Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33803 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1266 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)(19,20,25,29)(21,30,31,47)(22,27,24,39)(23,35,38,49)(26,41,42,33)(28,34,46,37)(32,48,40,45)(36,43,50,44);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(21,31)(26,42)(30,47)(32,40)(33,41)(36,50)(43,44)(45,48);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(19,21)(20,26)(22,32)(23,36)(24,40)(25,31)(27,43)(28,45)(29,42)(30,35)(33,37)(34,41)(38,50)(39,44)(46,48)(47,49);(4,10)(8,11)(9,18)(16,17)(19,22,25,24)(20,27,29,39)(21,32,31,40)(23,34,38,37)(26,43,42,44)(28,35,46,49)(30,48,47,45)(33,36,41,50)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530603);