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×#52857 |
| 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 | |
4262 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)(6,11,13,9)(7,10,15,12)(14,17,18,16)(19,20)(21,22)(23,27)(24,25)(26,29)(28,30)(31,32)(33,34)(38,46)(41,48)(44,50)(47,49);(4,5)(6,13)(10,12)(14,18)(20,25)(22,26)(27,32)(30,33)(35,36,39,42)(37,40,45,43)(38,41,46,48)(44,47,50,49);(3,5,8,4)(6,11,13,9)(7,12,15,10)(14,17,18,16)(19,21)(20,26)(22,25)(23,28)(24,29)(27,33)(30,32)(31,34)(35,36)(37,43)(38,41)(39,42)(40,45)(44,49)(46,48)(47,50);(3,6)(4,9)(5,11)(7,14)(8,13)(10,16)(12,17)(15,18)(19,22,24,26)(20,21,25,29)(23,30,31,33)(27,28,32,34)(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)(19,23,24,31)(20,27,25,32)(21,28,29,34)(22,30,26,33)(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, 7999857);