Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator | C4 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| prime radical | C23 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,15)(8,12)(9,21)(10,13)(11,17)(14,23)(16,29)(18,28)(19,27)(20,33)(22,25)(24,31)(26,32)(30,36)(34,35);(5,7,10,18)(6,11,13,25)(8,16,19,30)(9,22,23,17)(12,24,27,35)(14,28,21,15)(20,34,32,31)(26,36,33,29);(1,3,4,2)(5,8,10,19)(6,12,13,27)(7,16,18,30)(9,20,23,32)(11,24,25,35)(14,26,21,33)(15,29,28,36)(17,31,22,34);(6,13)(8,19)(11,25)(14,21)(15,28)(16,30)(20,32)(31,34)Polycyclic presentation:
2^7|a^2=f,b^2=g,c^2=fg,e^2=g,b^a=be,d^a=dg,e^a=eg,e^b=eg,d^c=dg44 conjugacy classes — view the full character table.
| maximal subgroup of | #12827, #12941, #12954, … |
| maximal normal subgroup of | #12827, #12941, #12954, … |
| maximal quotient of | #17856, #17880, #17897, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1682);