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×#61757 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6132 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)(19,20,25,29)(21,26,32,42)(22,33,35,43)(23,28,24,37)(27,38,44,34)(30,45,48,50)(31,40,36,41)(39,46,49,47);(3,5)(4,8)(6,12)(7,13)(9,15)(10,11)(14,18)(16,17)(19,21)(20,26)(22,30)(23,36)(24,31)(25,32)(27,39)(28,41)(29,42)(33,45)(34,47)(35,48)(37,40)(38,46)(43,50)(44,49);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(19,22,25,35)(20,27,29,44)(21,30,32,48)(23,34,24,38)(26,39,42,49)(28,43,37,33)(31,46,36,47)(40,45,41,50);(5,13)(8,11)(12,18)(15,17);(4,10)(8,11)(9,16)(15,17)(19,23,25,24)(20,28,29,37)(21,31,32,36)(22,34,35,38)(26,40,42,41)(27,43,44,33)(30,46,48,47)(39,45,49,50)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485123);