Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33752 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33798 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1298 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)(21,25)(22,26)(23,36)(24,29)(27,33)(28,34)(30,37)(31,46)(32,40)(35,42)(38,43)(39,44)(41,47)(45,49)(48,50);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21,24,32)(20,25,29,40)(22,30,35,45)(23,31,34,44)(26,37,42,49)(27,38,41,48)(28,39,36,46)(33,43,47,50);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(20,26)(22,33)(23,34)(25,37)(27,28)(29,42)(30,43)(31,44)(35,47)(36,41)(38,39)(40,49)(45,50)(46,48);(4,10)(8,11)(9,18)(16,17)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530591);