Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) = C2×D4×#3003 (order 2048) · Inn(G) = #121 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #121 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C23 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
98 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)(6,11)(7,14)(8,9)(12,19)(13,17)(15,16);(1,3,4,2)(5,6)(7,12)(8,17)(9,11)(10,13)(14,18)(15,20)(16,19);(5,7,8,15,10,16,9,14)(6,12,17,20,13,19,11,18)Polycyclic presentation:
2^7|a^2=f,b^2=f,c^2=eg,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg32 conjugacy classes — view the full character table.
| maximal subgroup of | #12940, #12941, #12945, … |
| maximal normal subgroup of | #12940, #12941, #12945, … |
| maximal quotient of | #9608, #9610, #9613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 684);