Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| isoclinism | isoclinic class of #64945 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5 ⋊ #400 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D5 |
| commutator, socle, prime radical, cosocle | C23×C5 | C23 |
| Fitting | C2×C5×C4⋊C4 | C2 |
| hypercenter | C2×C4⋊C4 | D5 |
| hypocenter | C5 | #400 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,4,5,3)(7,9)(18,20,21,24)(19,22,23,25);(10,12,13,17)(11,14,15,16)(20,24)(22,25);(2,3)(4,5)(6,7)(8,9)(10,11,13,15)(12,16,17,14)(18,19,21,23)(20,22,24,25)Polycyclic presentation:
2^6,5|a^2=e,b^2=ef,c^2=f,b^a=bef,c^a=cd,g^a=g^4,c^b=cf62 conjugacy classes — view the full character table.
| maximal subgroup of | #10597950, #10597987, #10598929, … |
| maximal normal subgroup of | #10597950, #10597987, #10598929, … |
| maximal quotient of | #10600208, #10600210, #10600216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 613);