Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
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 | |
138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,15,11,14,9,23)(7,12,18,27,17,28,24,26)(8,19,21,29,22,33,20,13)(16,30,32,25,31,35,34,36);(1,2,3,4)(5,7,9,18,11,17,10,24)(6,12,15,26,14,28,23,27)(8,16,20,32,22,31,21,34)(13,25,19,30,29,36,33,35);(5,8,10,21,11,22,9,20)(6,13,23,33,14,29,15,19)(7,16,24,34,17,31,18,32)(12,25,27,35,28,36,26,30)Polycyclic presentation:
2^7|a^2=f,b^2=dfg,c^2=eg,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,f^b=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #13019, #13020, #13023, … |
| maximal normal subgroup of | #13019, #13020, #13023, … |
| maximal quotient of | #9669, #9671, #9725, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 711);