Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#55840 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4202 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);(4,8)(5,11)(10,15)(13,17)(19,20)(21,24)(22,27)(23,28)(25,29)(26,31)(30,32)(33,34)(35,36,40,44)(37,42,38,41)(39,47,48,43)(45,49,46,50);(6,14)(10,15)(13,17)(16,18)(20,21)(22,29)(26,28)(30,34)(35,37,40,38)(36,41,44,42)(39,46,48,45)(43,50,47,49);(3,4,7,8)(5,12,11,9)(6,10,14,15)(13,18,17,16)(19,21,24,20)(22,27,29,25)(23,28,31,26)(30,33,34,32)(35,38,40,37)(36,42,44,41)(39,46,48,45)(43,50,47,49);(3,5,7,11)(4,9,8,12)(6,13,14,17)(10,16,15,18)(19,22)(20,25)(21,27)(23,30)(24,29)(26,32)(28,33)(31,34)(35,36)(37,42)(38,41)(39,43)(40,44)(45,50)(46,49)(47,48);(3,6)(4,10)(5,13)(7,14)(8,15)(9,16)(11,17)(12,18)(19,23)(20,26)(21,28)(22,30)(24,31)(25,32)(27,33)(29,34)(35,39)(36,43)(37,45)(38,46)(40,48)(41,49)(42,50)(44,47)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008896);