Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C23 |
| commutator | C22 | C23×C4 |
| Frattini, socle, cosocle | C23 | C24 |
| prime radical | C22×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
468 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 468 conjugacy classes of subgroups — too many to draw. Only its 188 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,4)(5,6,7,8)(9,10)(11,14)(12,13)(15,16);(1,2,3,4)(6,8);(10,13)(14,16);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^7|a^2=g,b^2=eg,d^2=f,b^a=be,c^a=cf56 conjugacy classes — view the full character table.
| derived subgroup of | #72227653, #72227655, #72227840, … |
| Frattini subgroup of | #94085, #94087, #94089, … |
| maximal subgroup of | #7563, #7578, #7597, … |
| maximal normal subgroup of | #7563, #7578, #7597, … |
| maximal quotient of | #13756, #13761, C4×#386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1029);