Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61790 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6218 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)(22,28)(23,29)(24,38)(25,30)(27,34)(31,40)(32,41)(33,43)(35,44)(36,42)(37,45)(39,47)(46,49)(48,50);(3,5)(4,8)(6,12)(7,13)(9,15)(10,11)(14,18)(16,17)(19,21,25,33)(20,26,30,43)(22,31,35,48)(23,36,24,32)(27,39,45,49)(28,40,44,50)(29,42,38,41)(34,47,37,46);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(19,22)(20,27)(21,31)(23,34)(24,37)(25,35)(26,39)(28,38)(29,44)(30,45)(32,46)(33,48)(36,47)(40,41)(42,50)(43,49);(5,13)(8,11)(12,18)(15,17)(19,22)(20,28)(21,31)(23,37)(24,34)(25,35)(26,40)(27,38)(29,45)(30,44)(32,47)(33,48)(36,46)(39,41)(42,49)(43,50);(4,10)(8,11)(9,16)(15,17)(19,23,25,24)(20,29,30,38)(21,32,33,36)(22,34,35,37)(26,41,43,42)(27,44,45,28)(31,46,48,47)(39,40,49,50)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485161);