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×#54721 |
| 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 | |
4246 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)(19,20,24,23)(21,27,28,25)(22,30,31,26)(29,32,34,33)(35,36)(37,40)(38,46)(39,42)(41,47)(43,45)(44,49)(48,50);(6,14)(10,15)(13,17)(16,18)(21,28)(25,27)(29,34)(32,33)(35,36,39,42)(37,43,45,40)(38,46,47,41)(44,48,50,49);(3,4,7,8)(5,12,11,9)(6,10,14,15)(13,18,17,16)(19,21,24,28)(20,25,23,27)(22,29,31,34)(26,32,30,33)(35,36)(37,43)(38,41)(39,42)(40,45)(44,49)(46,47)(48,50);(3,5,7,11)(4,9,8,12)(6,13,14,17)(10,16,15,18)(19,22,24,31)(20,26,23,30)(21,29,28,34)(25,32,27,33)(35,37,39,45)(36,40,42,43)(38,44,47,50)(41,48,46,49);(3,6,7,14)(4,10,8,15)(5,13,11,17)(9,16,12,18)(19,23,24,20)(21,27,28,25)(22,30,31,26)(29,33,34,32)(35,38,39,47)(36,41,42,46)(37,44,45,50)(40,48,43,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008816);