Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C22×#41 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
377 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 377 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,10,14)(7,15,17,20)(8,13,9,12)(11,18,19,16);(5,7)(6,11)(8,16)(9,18)(10,17)(12,15)(13,20)(14,19);(2,4);(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,15,19,20)Polycyclic presentation:
2^7|a^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg32 conjugacy classes — view the full character table.
| maximal subgroup of | #9345, #9361, #9364, … |
| maximal normal subgroup of | #9345, #9361, #9364, … |
| maximal quotient of | #18136, #18141, #18146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1739);