Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C5 |
No aliases for this group.
| Fitting | C52×#120 ⋊ C2 |
| Sylow | C52 ⋊ D42 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C5⋊D5 |
| commutator, socle, cosocle | C22×C52 | C24 |
| Fitting | C52×#120 | C2 |
| hypercenter | #120 | C5⋊D5 |
| hypocenter | C52 | D42 |
| LCS | G ▷ C22×C52 ▷ C52 | |
| UCS | C1 ◁ C22 ◁ #120 | |
1712 conjugacy classes of subgroups; only the 259 normal subgroups are shown.
This group has 1712 conjugacy classes of subgroups — too many to draw. Only its 259 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|b^a=bf,g^a=g^4,h^a=h^4,d^b=df,d^c=de193 conjugacy classes — view the full character table.
| maximal subgroup of | D42×C5⋊D5, #72221182, #72221186, … |
| maximal normal subgroup of | D42×C5⋊D5, #72221182, #72221186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 9471);