Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×#135 ⋊ C2 |
| Sylow | C5 ⋊ #509 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#686 |
| commutator, cosocle | C5×C8 | C23 |
| Frattini | C8 | C22×D5 |
| Fitting | C5×#135 | C2 |
| socle | C2×C5 | C2×D8 |
| hypercenter | #135 | D5 |
| hypocenter | C5 | #509 |
| LCS | G ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #135 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4,5,3)(6,7)(8,15)(9,18)(10,12)(11,14)(13,16)(17,20)(19,21);(7,12)(11,19)(13,18)(15,20);(2,3)(4,5)(7,11)(8,14)(9,16)(12,19)(13,20)(15,18)(17,21)Polycyclic presentation:
2^6,5|d^2=ef,e^2=f,c^a=cd,d^a=de,e^a=ef,g^a=g^4,c^b=cf,d^c=de,e^c=ef44 conjugacy classes — view the full character table.
| maximal subgroup of | #10600035, #10600036, #10600042, … |
| maximal normal subgroup of | #10600035, #10600036, #10600042, … |
| maximal quotient of | #10601442, #10601445, #10601446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 820);