Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | Aut(G) = #72218133 (order 2048) · Inn(G) = #120 · Out(G) = D42 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C24 | C23 |
| prime radical | #534 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
215 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 215 conjugacy classes of subgroups — too many to draw. Only its 46 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)(7,10)(8,11)(9,12)(13,16)(14,17)(15,19)(18,20);(5,7,9,14)(6,10,12,17)(8,13,15,20)(11,18,19,16);(2,4)(5,8)(6,11)(7,13)(9,15)(10,16)(12,19)(14,20)(17,18)Polycyclic presentation:
2^7|b^2=g,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^c=dg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7395, #12092, #12094, … |
| maximal normal subgroup of | #7395, #12092, #12094, … |
| maximal quotient of | #9918, #9919, #9920, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 754);