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×#56174 |
| 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 | |
4230 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)(6,11)(7,12)(9,13)(10,15)(14,16)(17,18)(19,20,24,22)(21,25,29,27)(23,30,31,26)(28,33,34,32)(35,36,40,44)(37,42,38,41)(39,47,48,43)(45,49,46,50);(4,5)(6,13)(10,12)(14,18)(20,22)(25,27)(26,30)(32,33)(35,37,40,38)(36,41,44,42)(39,46,48,45)(43,50,47,49);(3,5,8,4)(6,11,13,9)(7,12,15,10)(14,17,18,16)(19,21,24,29)(20,25,22,27)(23,28,31,34)(26,32,30,33)(35,38,40,37)(36,42,44,41)(39,46,48,45)(43,50,47,49);(3,6)(4,9)(5,11)(7,14)(8,13)(10,16)(12,17)(15,18)(19,22,24,20)(21,27,29,25)(23,30,31,26)(28,33,34,32)(35,36)(37,42)(38,41)(39,43)(40,44)(45,50)(46,49)(47,48);(3,7,8,15)(4,10,5,12)(6,14,13,18)(9,16,11,17)(19,23)(20,26)(21,28)(22,30)(24,31)(25,32)(27,33)(29,34)(35,39,40,48)(36,43,44,47)(37,45,38,46)(41,49,42,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007072);