Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C52 · Schur multiplier (H2) = C25×C5 |
| direct product | C52×#518 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C52 | C24 |
| commutator | C2 | C23×C4×C52 |
| Frattini, cosocle | C22 | C24×C52 |
| prime radical | C22×C52×D4 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C52 ◁ G | |
1352 conjugacy classes of subgroups; only the 968 normal subgroups are shown.
This group has 1352 conjugacy classes of subgroups — too many to draw. Only its 968 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) = 400 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2,3,4,5)(11,12,13,14)(15,16)(17,21)(18,19)(20,22);(6,7,8,9,10)(16,19)(20,21);(15,16)(17,20)(18,19)(21,22);(15,17)(16,20)(18,22)(19,21)850 conjugacy classes — view the full character table.
| maximal subgroup of | C5⋊D5×#518, C52×#3484, #72223221, … |
| maximal normal subgroup of | C5⋊D5×#518, C52×#3484, #72223221, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 9728);