Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C52 ⋊ #340 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C23 | D52⋊C2 |
| commutator | #1039 | C42 |
| Fitting, socle, prime radical | C23×C52 | D4 |
| cosocle | C22×#1039 | C22 |
| hypocenter | C52 | #340 |
| LCS | G ▷ #1039 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C23 | |
244 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 53 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=c,b^2=def,c^2=f,b^a=bcd,g^a=g^2,h^a=h^3,c^b=cf,g^b=h^4,h^b=g^4,g^c=g^4,h^c=h^4100 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 2359);