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×#54470 |
| 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 | |
4284 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);(3,4,7,8)(5,11,12,9)(6,14,15,10)(13,16,18,17)(19,20)(21,27)(22,29)(23,31)(24,25)(26,32)(28,30)(33,34)(35,36)(37,41)(38,45)(39,42)(40,43)(44,48)(46,50)(47,49);(4,8)(5,12)(10,14)(13,18)(20,25)(22,27)(23,32)(28,34)(35,37,40,45)(36,41,43,38)(39,44,47,50)(42,48,49,46);(3,5,7,12)(4,9,8,11)(6,13,15,18)(10,16,14,17)(19,21,24,29)(20,22,25,27)(23,28,32,34)(26,30,31,33)(35,37)(36,38)(39,44)(40,45)(41,43)(42,46)(47,50)(48,49);(3,4)(5,9)(6,10)(7,8)(11,12)(13,16)(14,15)(17,18)(19,22)(20,21)(23,30)(24,27)(25,29)(26,28)(31,34)(32,33)(35,38)(36,37)(39,46)(40,41)(42,44)(43,45)(47,48)(49,50);(3,6)(4,10)(5,13)(7,15)(8,14)(9,16)(11,17)(12,18)(19,23)(20,26)(21,28)(22,30)(24,32)(25,31)(27,33)(29,34)(35,39,40,47)(36,42,43,49)(37,44,45,50)(38,46,41,48)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009075);