Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C52 ⋊ C2×M32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C2×D52 |
| commutator, socle | C2×C52 | C22×C8 |
| Fitting | C2×C52×C8 | C22 |
| prime radical | C22×C5⋊D5 | C8 |
| cosocle | C52×C8 | C23 |
| hypercenter | C2×C8 | D52 |
| hypocenter | C52 | C2×M32 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C8 ◁ C2×C8 | |
256 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 74 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|a^2=d,b^2=d,d^2=e,e^2=f,c^a=cf,g^a=g^4,c^b=cf,h^b=h^4184 conjugacy classes — view the full character table.
| maximal subgroup of | #72220684, #72220685, #72220686, … |
| maximal normal subgroup of | #72220684, #72220685, #72220686, … |
| maximal quotient of | #72242157 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 3158);