Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| Fitting | C32×H(5,𝔽2) ⋊ C22 |
| Sylow | C32 ⋊ H(7,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C24×D32 |
| commutator, socle, cosocle | C2×C32 | C26 |
| Fitting | C32×H(5,𝔽2) | C22 |
| hypercenter | H(5,𝔽2) | D32 |
| hypocenter | C32 | H(7,𝔽2) |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ H(5,𝔽2) | |
12389 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^7,3^2|b^a=bg,f^a=fg,e^b=eg,d^c=dg,i^e=hi^2,h^f=h^2,i^f=h^2i150 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153292);