Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19086 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19092 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3219 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2656 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);(4,9)(5,12)(6,7)(11,14)(13,15)(16,18)(19,20)(21,29)(22,26)(23,35)(24,27)(25,32)(28,38)(30,46)(31,43)(33,44)(34,41)(36,40)(37,42)(39,48)(45,47)(49,50);(3,4)(5,10)(6,14)(7,9)(8,11)(12,18)(13,17)(15,16)(19,21,24,31)(20,25,27,40)(22,30,34,47)(23,36,38,32)(26,39,41,50)(28,43,35,29)(33,49,37,48)(42,45,44,46);(3,5,8,13)(4,10,11,17)(6,12,7,15)(9,16,14,18)(19,22,24,34)(20,26,27,41)(21,30,31,47)(23,37,38,33)(25,39,40,50)(28,44,35,42)(29,45,43,46)(32,49,36,48);(4,11)(9,14)(10,17)(16,18)(20,27)(21,31)(26,41)(28,35)(30,47)(32,36)(42,44)(48,49)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258907);