Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C5 ⋊ #541 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, cosocle | C22×C5 | C24 |
| Fitting | C5×#126 | C2 |
| prime radical | C23×C5 | C23 |
| hypercenter | #126 | D5 |
| hypocenter | C5 | #541 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ #126 | |
192 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,4,5,3)(8,13)(11,12)(14,17,18,26)(15,20,21,24)(16,23,22,25)(19,28,27,29);(6,9,10,7)(8,12,13,11)(16,22)(19,27)(23,25)(28,29);(6,8)(7,11)(9,12)(10,13)(14,16,18,22)(15,19,21,27)(17,25,26,23)(20,29,24,28);(2,3)(4,5)(6,7,10,9)(8,11,13,12)(14,15,18,21)(16,19,22,27)(17,24,26,20)(23,29,25,28)Polycyclic presentation:
2^6,5|a^2=f,b^2=e,c^2=ef,d^2=e,d^a=de,g^a=g^4,c^b=ce,d^b=df50 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#66035, #10618011, #10618017, … |
| maximal normal subgroup of | C2×#66035, #10618011, #10618017, … |
| maximal quotient of | #10605991, #10605992, #10605998, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1373);