Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C52 ⋊ #454 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11717094 |
| commutator, cosocle | C2×C4×C52 | C23 |
| Frattini | C2×C4 | C2×D52 |
| Fitting | C52×#41 | C22 |
| socle | C2×C52 | #120 |
| hypercenter | #41 | D52 |
| hypocenter | C52 | #454 |
| LCS | G ▷ C2×C4×C52 ▷ C2×C52 ▷ C52 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #41 | |
336 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 64 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|c^2=f,d^2=f,e^2=f,b^a=be,e^a=ef,g^a=g^4,c^b=cd,d^b=df,e^b=ef,h^b=h^4,d^c=df112 conjugacy classes — view the full character table.
| maximal subgroup of | #72221241, #72223377, #72223386, … |
| maximal normal subgroup of | #72221241, #72223377, #72223386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 4216);