Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C52 · Schur multiplier (H2) = C2×C4×C5 |
| direct product | C52×#469 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C52 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C23×C52 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C52 ◁ C22×C4×C52 ◁ G | |
520 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 800 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3,4,5)(11,12)(13,14)(16,19)(17,20)(21,22);(6,7,8,9,10)(15,16,20,22,18,21,17,19);(12,14)(16,21)(19,22)400 conjugacy classes — view the full character table.
| Fitting subgroup of | C5×#10612597, C5×#10612596, #72242560, … |
| maximal subgroup of | C52×#3184, C5⋊D5×#469, #72223075, … |
| maximal normal subgroup of | C52×#3184, C5⋊D5×#469, #72223075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 5558);