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×#54703 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4392 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)(37,44)(38,41)(39,47)(40,43)(42,48)(45,49)(46,50);(6,14)(10,15)(13,17)(16,18)(21,28)(25,27)(29,34)(32,33)(35,37,40,38)(36,41,43,44)(39,45,48,46)(42,49,47,50);(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)(35,38,40,37)(36,41,43,44)(39,46,48,45)(42,49,47,50);(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)(35,36)(37,44)(38,41)(39,42)(40,43)(45,50)(46,49)(47,48);(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)(35,39)(36,42)(37,45)(38,46)(40,48)(41,49)(43,47)(44,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8010119);