Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,15,19,16,18,20,17);(1,2,3,4)(6,9)(10,11)(14,17)(15,20)(18,19);(5,7,8,12)(6,10,9,11)(14,18)(17,19)Polycyclic presentation:
2^7|a^2=f,b^2=def,c^2=eg,d^2=g,f^2=g,b^a=bd,c^a=ce,d^b=dg,f^b=fg38 conjugacy classes — view the full character table.
| maximal subgroup of | #8512, #8513, #10509, … |
| maximal normal subgroup of | #8512, #8513, #10509, … |
| maximal quotient of | #9866, #9867, #9900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 721);