Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53383 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(5,11)(9,14)(12,17)(16,18)(19,20,23,26)(21,27,28,24)(22,25,30,32)(29,33,34,31)(35,36,41,39)(37,44,38,42)(40,47,48,43)(45,49,46,50);(4,8)(6,13)(9,14)(12,17)(21,28)(24,27)(29,34)(31,33)(35,37,41,38)(36,42,39,44)(40,45,48,46)(43,49,47,50);(3,4,7,8)(5,9,11,14)(6,10,13,15)(12,16,17,18)(19,21)(20,24)(22,29)(23,28)(25,31)(26,27)(30,34)(32,33)(35,38,41,37)(36,42,39,44)(40,46,48,45)(43,49,47,50);(3,5)(4,9)(6,12)(7,11)(8,14)(10,16)(13,17)(15,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,34)(35,39,41,36)(37,42,38,44)(40,47,48,43)(45,49,46,50);(3,6)(4,10)(5,12)(7,13)(8,15)(9,16)(11,17)(14,18)(19,22,23,30)(20,25,26,32)(21,29,28,34)(24,31,27,33)(35,40,41,48)(36,43,39,47)(37,45,38,46)(42,49,44,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013109);