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×#54315 |
| 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);(3,4,8,5)(6,9,13,11)(7,12,15,10)(14,17,18,16)(19,20)(21,25)(22,29)(23,24)(26,30)(27,33)(28,31)(32,34)(35,36)(37,42)(38,44)(39,41)(40,43)(45,49)(46,50)(47,48);(4,5)(6,13)(7,15)(16,17)(20,24)(25,31)(26,29)(32,33)(35,37,41,38)(36,42,39,44)(40,46,48,45)(43,50,47,49);(3,5,8,4)(6,9,13,11)(7,12,15,10)(14,16,18,17)(19,21,23,28)(20,25,24,31)(22,27,30,34)(26,32,29,33)(35,38,41,37)(36,42,39,44)(40,46,48,45)(43,49,47,50);(3,6)(4,9)(5,11)(7,14)(8,13)(10,16)(12,17)(15,18)(19,20)(21,25)(22,26)(23,24)(27,32)(28,31)(29,30)(33,34)(35,39,41,36)(37,44,38,42)(40,47,48,43)(45,50,46,49);(3,7,8,15)(4,10,5,12)(6,14,13,18)(9,16,11,17)(19,22,23,30)(20,26,24,29)(21,27,28,34)(25,32,31,33)(35,40)(36,43)(37,45)(38,46)(39,47)(41,48)(42,49)(44,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010118);