Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#22257 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×C4×D4 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 50 points:
(1,2);(3,4)(5,12)(6,10)(7,16)(8,11)(9,14)(13,18)(15,17)(19,20,24,28,25,29,23,38)(21,26,32,40,33,41,31,47)(22,34,37,42,35,48,36,27)(30,43,46,49,44,50,45,39);(3,5)(4,9)(6,15)(7,13)(8,14)(10,18)(11,12)(16,17)(19,21,24,32,25,33,23,31)(20,26,28,40,29,41,38,47)(22,30,37,46,35,44,36,45)(27,39,34,43,42,49,48,50);(3,6,8,7)(4,10,11,16)(5,13,14,15)(9,17,12,18)(19,22,23,36,25,35,24,37)(20,27,28,34,29,42,38,48)(21,30,31,45,33,44,32,46)(26,39,40,43,41,49,47,50);(3,7,8,6)(4,10,11,16)(5,13,14,15)(9,18,12,17)(22,35)(27,42)(30,44)(34,48)(36,37)(39,49)(43,50)(45,46)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283577);