Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#2229 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C11 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C11 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C42×C11 ◁ C4×C8×C11 ◁ G | |
124 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 1408 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,15,17)(14,18,19,16)(20,21,23,26)(22,30,31,48)(24,36,34,49)(25,39,35,29)(27,43,41,32)(28,45,42,38)(33,50,40,46)(37,47,51,44);(12,14,15,19)(13,16,17,18)(21,26)(27,41)(28,42)(29,39)(30,48)(36,49)(44,47)(46,50);(20,22,23,31)(21,27,26,41)(24,37,34,51)(25,40,35,33)(28,46,42,50)(29,47,39,44)(30,49,48,36)(32,45,43,38)319 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 961);