Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) = #6437575 (order 512) · Inn(G) = #117 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #117 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C2×C8 | C23 |
| socle | C2 | #432 |
| prime radical | C23 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #97 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3,8,4,9,11,16,5,6,10,13,12,7,14,15);(2,6)(3,10)(7,9)(11,14);(2,7,6,9)(3,10)(8,16,13,15)(11,14)Polycyclic presentation:
2^7|a^2=e,c^2=d,e^2=f,f^2=g,b^a=bd,c^a=cf,d^a=dg,e^b=eg,e^c=eg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7225, #7226, #11521, … |
| maximal normal subgroup of | #7225, #7226, #11521, … |
| maximal quotient of | #12595, #12607, #12659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 897);