Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C52 · Schur multiplier (H2) = C24×C5 |
| direct product | C52×B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C52 | #120 |
| commutator, Frattini, cosocle | C23 | C23×C52 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C52 ◁ C23×C52 ◁ G | |
792 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 792 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) = 18 |
| transitive degree | μt(G) = 200 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2,3,4,5)(15,18);(6,7,8,9,10)(11,12)(13,15)(14,16)(17,18);(11,13)(12,15)(14,17)(16,18)400 conjugacy classes — view the full character table.
| Fitting subgroup of | C5×D5×B(4,𝔽2), C5×AGL(1,𝔽5)×B(4,𝔽2) |
| maximal subgroup of | C5×D5×B(4,𝔽2), B(4,𝔽2)×5−1+2 |
| maximal normal subgroup of | C5×D5×B(4,𝔽2), B(4,𝔽2)×5−1+2 |
| maximal quotient of | B(4,𝔽2)×5−1+2 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 5546);