Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C25 · Schur multiplier (H2) = C24 |
| direct product | C25×B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C25 | #120 |
| commutator | C23 | C23×C25 |
| Frattini, cosocle | C23×C5 | C23×C5 |
| socle | C2×C5 | C5×#120 |
| prime radical | C5×B(4,𝔽2) | C5 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C25 ◁ C23×C25 ◁ G | |
297 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 297 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 200 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25)(30,33);(26,27)(28,30)(29,31)(32,33);(26,28)(27,30)(29,32)(31,33)400 conjugacy classes — view the full character table.
| Fitting subgroup of | D25×B(4,𝔽2), B(4,𝔽2)×#1035 |
| maximal subgroup of | D25×B(4,𝔽2), B(4,𝔽2)×5−1+2 |
| maximal normal subgroup of | D25×B(4,𝔽2), B(4,𝔽2)×5−1+2 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 958);