Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61750 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(3,4)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20)(21,26)(22,34)(23,28)(24,37)(25,29)(27,35)(30,44)(31,40)(32,41)(33,47)(36,42)(38,43)(39,45)(46,49)(48,50);(3,5)(4,8)(6,12)(7,13)(9,15)(10,11)(14,18)(16,17)(19,21)(20,26)(22,30)(23,31)(24,33)(25,32)(27,39)(28,40)(29,41)(34,44)(35,45)(36,46)(37,47)(38,48)(42,49)(43,50);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(19,22,25,36)(20,27,29,43)(21,30,32,46)(23,35,24,38)(26,39,41,50)(28,42,37,34)(31,45,33,48)(40,49,47,44);(5,13)(8,11)(12,18)(15,17)(19,23,25,24)(20,28,29,37)(21,31,32,33)(22,35,36,38)(26,40,41,47)(27,42,43,34)(30,45,46,48)(39,49,50,44);(4,10)(8,11)(9,16)(15,17)(21,32)(26,41)(30,46)(31,33)(39,50)(40,47)(44,49)(45,48)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485118);