Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C52 ⋊ #322 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C42 | D52 |
| commutator | C2×C52 | C4×C8 |
| Fitting, cosocle | C42×C52 | C22 |
| socle, prime radical | C22×C52 | C42 |
| hypocenter | C52 | #322 |
| LCS | G ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C42 | |
176 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 320 |
Polycyclic presentation:
2^6,5^2|a^2=d,b^2=c,c^2=e,d^2=f,b^a=bf,h^a=h^4,g^b=g^4232 conjugacy classes — view the full character table.
| maximal subgroup of | #72242174, #72242277, #72242284 |
| maximal normal subgroup of | #72242174, #72242277, #72242284 |
| maximal quotient of | #72242171 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 1600);