Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) = #72236620 (order 2048) · Inn(G) = #121 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
138 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 3 generators on 20 points:
(1,2)(3,4)(5,6,9,13)(7,14,15,11)(8,10,17,12)(16,19,20,18);(6,10)(7,15)(8,17)(11,19)(12,13)(14,18);(2,4)(5,7,8,16,9,15,17,20)(6,11,12,19,13,14,10,18)Polycyclic presentation:
2^7|a^2=g,c^2=d,d^2=g,b^a=bd,c^a=ce,d^a=dg,c^b=cf,d^b=dg,e^c=eg,f^c=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2046, #23520, #23521, … |
| maximal normal subgroup of | C2×#2046, #23520, #23521, … |
| maximal quotient of | #10442, #10443, #10444, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 778);