Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52851 |
| 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 | |
4278 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)(5,8)(6,9)(7,12)(10,15)(11,13)(14,17)(16,18)(19,20,23,24)(21,25,29,27)(22,30,31,26)(28,33,34,32)(38,46)(41,48)(44,50)(47,49);(4,5)(6,13)(10,12)(14,18)(20,24)(25,27)(26,30)(32,33)(35,36,39,42)(37,40,45,43)(38,41,46,48)(44,47,50,49);(3,5,8,4)(6,9,13,11)(7,12,15,10)(14,16,18,17)(19,21)(20,25)(22,28)(23,29)(24,27)(26,32)(30,33)(31,34)(35,36)(37,43)(38,41)(39,42)(40,45)(44,49)(46,48)(47,50);(3,6,8,13)(4,9,5,11)(7,14,15,18)(10,16,12,17)(19,20)(21,27)(22,26)(23,24)(25,29)(28,33)(30,31)(32,34)(35,37)(36,40)(38,44)(39,45)(41,47)(42,43)(46,50)(48,49);(3,7,8,15)(4,10,5,12)(6,14,13,18)(9,16,11,17)(19,22,23,31)(20,26,24,30)(21,28,29,34)(25,32,27,33)(35,38)(36,41)(37,44)(39,46)(40,47)(42,48)(43,49)(45,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999888);