Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) = #72231459 (order 2048) · Inn(G) = C2×D8 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D8 |
| commutator | C8 | C22×C4 |
| Frattini, cosocle | C2×C8 | C23 |
| prime radical | C2×D16 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4,3)(5,6,8,11)(7,10,16,18)(9,17,15,13)(12,20,19,14);(6,10)(7,14)(9,15)(11,18)(12,13)(16,20)(17,19);(1,3,4,2)(6,11)(10,18)(12,19)(13,17)Polycyclic presentation:
2^7|a^2=e,c^2=eg,d^2=fg,f^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fg,d^b=df,f^b=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7390, #12067, #12078, … |
| maximal normal subgroup of | #7390, #12067, #12078, … |
| maximal quotient of | #12045, #12073, #12086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 909);