Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C52 ⋊ #533 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D52 |
| commutator, socle, cosocle | C22×C52 | C24 |
| Fitting | C42×C52 | C22 |
| prime radical | C22×C5⋊D5 | C23 |
| hypercenter | C42 | D52 |
| hypocenter | C52 | #533 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ C42 | |
552 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^6,5^2|a^2=f,b^2=f,c^2=e,d^2=f,b^a=bf,c^a=cf,d^a=de,g^a=g^4,c^b=cf,d^b=de,h^b=h^4148 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 7430);