Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C32 ⋊ #2021 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #63983 |
| commutator, cosocle | C24×C32 | C23 |
| Frattini | C24 | C2×D32 |
| Fitting | C25×C32 | C22 |
| socle | C22×C32 | #120 |
| hypercenter | C25 | D32 |
| hypocenter | C32 | #2021 |
| LCS | G ▷ C24×C32 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C25 | |
1529 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 1529 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^7,3^2|b^a=bd,c^a=ce,f^a=fg,i^a=hi^2,c^b=cf,d^c=dg,h^c=h^2,i^c=h^2i114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 26815);