Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60859 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| socle | C23 | C23×D4 |
| prime radical | C2×#3455 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(3,4)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20)(21,26)(22,34)(23,28)(24,37)(25,29)(27,35)(30,44)(31,40)(32,41)(33,47)(36,42)(38,43)(39,46)(45,50)(48,49);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21)(20,26)(22,30)(23,32)(24,33)(25,31)(27,39)(28,41)(29,40)(34,44)(35,46)(36,45)(37,47)(38,48)(42,50)(43,49);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(19,22)(20,27)(21,30)(23,35)(24,38)(25,36)(26,39)(28,42)(29,43)(31,45)(32,46)(33,48)(34,37)(40,49)(41,50)(44,47);(4,10)(8,11)(9,18)(16,17)(21,31)(26,40)(30,45)(32,33)(39,49)(41,47)(44,50)(46,48);(19,23,25,24)(20,28,29,37)(21,32,31,33)(22,35,36,38)(26,41,40,47)(27,42,43,34)(30,46,45,48)(39,50,49,44)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482895);