Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×SD32 ⋊ C2 |
| Sylow | C5 ⋊ #508 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D5×D8 |
| commutator, cosocle | C5×C8 | C23 |
| Frattini | C8 | C22×D5 |
| Fitting | C5×SD32 | C2 |
| socle | C2×C5 | C2×D8 |
| hypercenter | SD32 | D5 |
| hypocenter | C5 | #508 |
| LCS | G ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ SD32 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,4,5,3)(6,8,11,19)(7,13,12,16)(9,21,23,36)(10,24,22,18)(14,31,25,37)(15,33,26,29)(17,34,35,20)(27,32,28,30);(7,12)(8,17)(9,20)(10,22)(13,27)(14,30)(16,28)(18,36)(19,35)(21,24)(23,34)(25,32)(29,31)(33,37);(2,3)(4,5)(6,7)(8,16)(9,14)(10,15)(11,12)(13,19)(17,27)(18,33)(20,32)(21,37)(22,26)(23,25)(24,29)(28,35)(30,34)(31,36)Polycyclic presentation:
2^6,5|c^2=f,d^2=ef,e^2=f,b^a=bf,c^a=cf,g^a=g^4,c^b=cd,d^b=de,e^b=ef,d^c=de,e^c=ef44 conjugacy classes — view the full character table.
| maximal subgroup of | #10604128, #10604129, C2×#65205, … |
| maximal normal subgroup of | #10604128, #10604129, C2×#65205, … |
| maximal quotient of | #10599967, #10599974, #10599975, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 543);