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×#56865 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4116 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)(5,11,13,9)(6,14,15,10)(12,16,18,17)(19,20,24,27)(21,23,26,32)(22,25,31,34)(28,30,29,33)(35,36)(37,40)(38,46)(39,42)(41,47)(43,45)(44,49)(48,50);(4,8)(6,15)(9,11)(12,18)(21,26)(23,32)(28,29)(30,33)(35,36,39,42)(37,43,45,40)(38,46,47,41)(44,48,50,49);(3,5,7,13)(4,9,8,11)(6,12,15,18)(10,16,14,17)(19,21,24,26)(20,23,27,32)(22,29,31,28)(25,33,34,30)(35,36)(37,43)(38,41)(39,42)(40,45)(44,49)(46,47)(48,50);(3,4)(5,11)(6,10)(7,8)(9,13)(12,17)(14,15)(16,18)(19,22)(20,25)(21,28)(23,30)(24,31)(26,29)(27,34)(32,33)(35,37)(36,40)(38,44)(39,45)(41,48)(42,43)(46,49)(47,50);(3,6,7,15)(4,10,8,14)(5,12,13,18)(9,16,11,17)(19,23)(20,26)(21,27)(22,30)(24,32)(25,29)(28,34)(31,33)(35,38)(36,41)(37,44)(39,47)(40,48)(42,46)(43,49)(45,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007446);