Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C52 ⋊ #568 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×D52 |
| commutator, socle | C2×C52 | C23×C4 |
| Fitting | C52×#41 | C22 |
| prime radical | #41×C5⋊D5 | C2 |
| cosocle | C4×C52 | C24 |
| hypercenter | #41 | D52 |
| hypocenter | C52 | #568 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C4 ◁ #41 | |
680 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 680 conjugacy classes of subgroups — too many to draw. Only its 192 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=e,b^2=e,e^2=f,d^a=df,g^a=g^4,c^b=cf,h^b=h^4,d^c=df154 conjugacy classes — view the full character table.
| maximal subgroup of | #72221063, #72223278, #72223431, … |
| maximal normal subgroup of | #72221063, #72223278, #72223431, … |
| maximal quotient of | #72242678, #72242679 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 8886);