Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60882 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3512 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6050 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)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20)(21,26)(22,28)(23,29)(24,38)(25,30)(27,35)(31,40)(32,41)(33,48)(34,42)(36,43)(37,44)(39,45)(46,49)(47,50);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21,25,34)(20,26,30,42)(22,31,36,46)(23,32,24,33)(27,39,44,50)(28,40,43,49)(29,41,38,48)(35,45,37,47);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(19,22)(20,27)(21,31)(23,35)(24,37)(25,36)(26,39)(28,38)(29,43)(30,44)(32,45)(33,47)(34,46)(40,48)(41,49)(42,50);(4,10)(8,11)(9,18)(16,17)(19,22)(20,28)(21,31)(23,37)(24,35)(25,36)(26,40)(27,38)(29,44)(30,43)(32,47)(33,45)(34,46)(39,48)(41,50)(42,49);(19,23,25,24)(20,29,30,38)(21,32,34,33)(22,35,36,37)(26,41,42,48)(27,43,44,28)(31,45,46,47)(39,49,50,40)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482921);