Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C24 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
128 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,18,16,20,15,19);(1,2)(3,4)(6,9)(10,12);(2,4)(5,7,8,11)(6,10,9,12)(14,18)(15,17)(19,20)Polycyclic presentation:
2^7|a^2=eg,c^2=d,e^2=g,b^a=bd,c^a=ce,c^b=cf,e^c=eg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2077, #23638, #23640, … |
| maximal normal subgroup of | C2×#2077, #23638, #23640, … |
| maximal quotient of | #10619, #10620, #10625, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 809);