Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×D4×C9 ⋊ C2 |
| Sylow | C9 ⋊ H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C23×D9 |
| commutator, cosocle | C2×C9 | C24 |
| Frattini, socle | C2×C3 | C23×D3 |
| Fitting | C2×D4×C9 | C2 |
| hypercenter | C2×D4 | D9 |
| hypocenter | C9 | H(5,𝔽2) |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C2×D4 | |
249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,6,8,7,2,4,3,9,5)(11,15)(14,17);(12,16)(14,17);(10,12)(11,14)(13,16)(15,17);(2,4)(3,7)(5,6)(8,9)(10,11)(12,14)(13,15)(16,17)Polycyclic presentation:
2^5,3^2|f^3=g^2,d^a=de,f^a=f^2g,g^a=g^2,c^b=ce57 conjugacy classes — view the full character table.
| maximal subgroup of | #10588421, #10588422, #10588426, … |
| maximal normal subgroup of | #10588421, #10588422, #10588426, … |
| maximal quotient of | #10589585, #10589588, #10589592, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 358);