Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C9×#41 ⋊ 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 | C9×#41 | C2 |
| hypercenter | #41 | D9 |
| hypocenter | C9 | H(5,𝔽2) |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ #41 | |
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)(10,12,13,16)(11,15,14,17);(11,14)(15,17);(10,11)(12,15)(13,14)(16,17);(2,4)(3,7)(5,6)(8,9)(12,16)(15,17)Polycyclic presentation:
2^5,3^2|d^2=e,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 | #10588502, #10588505, #10588879, … |
| maximal normal subgroup of | #10588502, #10588505, #10588879, … |
| maximal quotient of | #10589621, #10589623, #10589625, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 363);