Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #625 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C32 ⋊ C22⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C4 | D32 |
| commutator | C2×C32 | C2×C8 |
| Fitting, cosocle | C2×C32×C4 | C22 |
| socle | C22×C32 | C2×C4 |
| prime radical | C22×C3⋊D3 | C4 |
| hypocenter | C32 | C22⋊C8 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C4 | |
127 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 18 points:
(2,4)(5,7,8,12,9,6,10,11)(14,15)(16,17,18);(1,2,3,4)(5,6,8,11,9,7,10,12)(13,14,15)(17,18)Polycyclic presentation:
2^5,3^2|a^2=c,b^2=cd,c^2=e,b^a=bd,g^a=g^2,f^b=f^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10590027, #10590028, #10590033, … |
| maximal normal subgroup of | #10590027, #10590028, #10590033, … |
| maximal quotient of | #10589082, #10589085, #10589108, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 205);