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×#57806 |
| 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);(3,4,7,10)(5,11,13,8)(6,9,14,16)(12,17,18,15)(19,20,24,25)(21,22,29,26)(23,31,32,27)(28,33,34,30)(36,40)(37,43)(38,46)(48,50);(6,14)(9,16)(12,18)(15,17)(20,25)(22,26)(23,32)(28,34)(35,36,39,40)(37,44,43,41)(38,42,46,47)(45,50,49,48);(3,5,7,13)(4,8,10,11)(6,12,14,18)(9,15,16,17)(19,21,24,29)(20,26,25,22)(23,28,32,34)(27,33,31,30)(35,36)(37,41)(38,42)(39,40)(43,44)(45,48)(46,47)(49,50);(3,4)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,22,24,26)(20,21,25,29)(23,30,32,33)(27,28,31,34)(35,37,39,43)(36,41,40,44)(38,45,46,49)(42,48,47,50);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(19,23)(20,27)(21,28)(22,30)(24,32)(25,31)(26,33)(29,34)(35,38)(36,42)(37,45)(39,46)(40,47)(41,48)(43,49)(44,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008737);