Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60753 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60869 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
5990 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)(19,20,25,30)(21,26,32,40)(22,35,37,44)(23,29,24,28)(27,38,43,36)(31,45,46,50)(33,42,34,41)(39,48,49,47);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21)(20,26)(22,31)(23,33)(24,34)(25,32)(27,39)(28,41)(29,42)(30,40)(35,45)(36,47)(37,46)(38,48)(43,49)(44,50);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(19,22)(20,27)(21,31)(23,36)(24,38)(25,37)(26,39)(28,35)(29,44)(30,43)(32,46)(33,47)(34,48)(40,49)(41,45)(42,50);(4,10)(8,11)(9,18)(16,17)(21,32)(26,40)(31,46)(33,34)(39,49)(41,42)(45,50)(47,48);(19,23,25,24)(20,28,30,29)(21,33,32,34)(22,36,37,38)(26,41,40,42)(27,35,43,44)(31,47,46,48)(39,45,49,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482906);