Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
248 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 3 generators on 16 points:
(1,2)(3,4)(9,10)(11,13)(12,15)(14,16);(5,6)(7,8)(10,13)(11,16)(14,15);(2,4)(6,8)Polycyclic presentation:
2^7|d^2=g,b^a=bd,c^a=ce,d^a=dg,c^b=cf,d^b=dg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7076, #7084, #12102, … |
| maximal normal subgroup of | #7076, #7084, #12102, … |
| maximal quotient of | #9914, #9915, #9918, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 731);