Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C5 |
No aliases for this group.
| Fitting | C52×#135 ⋊ C2 |
| Sylow | C52 ⋊ #575 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×C5⋊D5 |
| commutator, cosocle | C4×C52 | C24 |
| Frattini | C4 | C23×C5⋊D5 |
| Fitting | C52×#135 | C2 |
| socle | C2×C52 | C22×D4 |
| hypercenter | #135 | C5⋊D5 |
| hypocenter | C52 | #575 |
| LCS | G ▷ C4×C52 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #135 | |
1048 conjugacy classes of subgroups; only the 219 normal subgroups are shown.
This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 219 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 400 |
Polycyclic presentation:
2^6,5^2|e^2=f,d^a=df,g^a=g^4,h^a=h^4,c^b=cf,d^b=df,d^c=de,e^c=ef,e^d=ef190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 9648);