Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C52 ⋊ #508 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D5×#686 |
| commutator, cosocle | C52×C8 | C23 |
| Frattini | C8 | C2×D52 |
| Fitting | C52×Q16 | C22 |
| socle | C2×C52 | C2×D8 |
| hypercenter | Q16 | D52 |
| hypocenter | C52 | #508 |
| LCS | G ▷ C52×C8 ▷ C4×C52 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16 | |
192 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 48 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|c^2=f,d^2=ef,e^2=f,b^a=bf,c^a=cf,g^a=g^4,c^b=cd,d^b=de,e^b=ef,h^b=h^4,d^c=de,e^c=ef100 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 3579);