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×#53328 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(5,11)(9,14)(12,17)(16,18)(19,20)(21,27)(22,30)(23,24)(25,28)(26,31)(29,32)(33,34)(35,36,39,40)(37,43,44,41)(38,46,47,42)(45,48,50,49);(4,8)(6,13)(9,14)(12,17)(20,24)(21,28)(22,31)(32,33)(36,40)(37,44)(42,46)(45,50);(3,4,7,8)(5,9,11,14)(6,10,13,15)(12,16,17,18)(19,20)(21,25)(22,26)(23,24)(27,28)(29,32)(30,31)(33,34)(35,36)(37,41)(38,42)(39,40)(43,44)(45,48)(46,47)(49,50);(3,5)(4,9)(6,12)(7,11)(8,14)(10,16)(13,17)(15,18)(19,21)(20,25)(22,29)(23,28)(24,27)(26,32)(30,33)(31,34)(35,37,39,44)(36,41,40,43)(38,45,47,50)(42,48,46,49);(3,6)(4,10)(5,12)(7,13)(8,15)(9,16)(11,17)(14,18)(19,22,23,31)(20,26,24,30)(21,29,28,34)(25,32,27,33)(35,38,39,47)(36,42,40,46)(37,45,44,50)(41,48,43,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013110);