Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ #2262 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10591025 |
| commutator, cosocle | C32×C16 | C23 |
| Frattini | C16 | C2×D32 |
| Fitting | C32×Q32 | C22 |
| socle | C2×C32 | C2×D16 |
| hypercenter | Q32 | D32 |
| hypocenter | C32 | #2262 |
| LCS | G ▷ C32×C16 ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ Q32 | |
254 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^7,3^2|c^2=g,d^2=ef,e^2=fg,f^2=g,b^a=bd,d^a=de,e^a=ef,f^a=fg,h^a=h^2,i^a=i^2,d^b=de,e^b=ef,f^b=fg,i^b=hi^2,h^c=h^2,i^c=i^287 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 34585);