Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C52×C8 ⋊ C22 |
| Sylow | C52 ⋊ #576 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D5×D20 |
| commutator, cosocle | C4×C52 | C24 |
| Frattini | C4 | C22×D52 |
| Fitting | C2×C52×C8 | C22 |
| socle | C2×C52 | C22×D4 |
| hypercenter | C2×C8 | D52 |
| hypocenter | C52 | #576 |
| LCS | G ▷ C4×C52 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 | |
612 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 80 |
Polycyclic presentation:
2^6,5^2|c^2=f,d^2=ef,e^2=f,c^a=cf,g^a=g^4,d^b=de,e^b=ef,h^b=h^4154 conjugacy classes — view the full character table.
| maximal subgroup of | #72220877, #72221382, #72221389, … |
| maximal normal subgroup of | #72220877, #72221382, #72221389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 8123);