Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×M32 ⋊ C2 |
| Sylow | C5 ⋊ #504 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C22×D5 |
| commutator, socle | C2×C5 | C22×C8 |
| Fitting | C5×M32 | C2 |
| prime radical | #709 | C4 |
| cosocle | C5×C8 | C23 |
| hypercenter | M32 | D5 |
| hypocenter | C5 | #504 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C8 ◁ M32 | |
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)(6,8,9,17,10,18,19,30,11,16,20,28,21,29,31,36)(7,13,14,25,15,26,27,35,12,22,23,32,24,33,34,37);(7,12)(8,16)(14,23)(15,24)(17,28)(18,29)(27,34)(30,36);(2,3)(4,5)(6,7)(8,13)(9,14)(10,15)(11,12)(16,22)(17,25)(18,26)(19,27)(20,23)(21,24)(28,32)(29,33)(30,35)(31,34)(36,37)Polycyclic presentation:
2^6,5|c^2=d,d^2=e,e^2=f,b^a=bf,g^a=g^4,c^b=cf80 conjugacy classes — view the full character table.
| maximal subgroup of | #10598669, #10599915, #10599916, … |
| maximal normal subgroup of | #10598669, #10599915, #10599916, … |
| maximal quotient of | #10599861, #10599865, #10599867, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 534);