Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) = #72252166 (order 4096) · Inn(G) = #121 · Out(G) = #2408 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C23 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,14)(7,15,18,30)(8,19,22,12)(9,23,25,13)(11,24,27,17)(16,32,20,31)(21,29,36,34)(26,33,28,35);(5,7,10,18)(6,11,14,27)(8,20,22,16)(9,24,25,17)(12,28,19,26)(13,30,23,15)(21,33,36,35)(29,31,34,32);(2,4)(5,8,9,21,10,22,25,36)(6,12,13,29,14,19,23,34)(7,16,17,33,18,20,24,35)(11,26,15,31,27,28,30,32)Polycyclic presentation:
2^7|a^2=g,b^2=g,c^2=d,d^2=g,b^a=bd,c^a=ce,d^a=dg,c^b=cf,d^b=dg,e^c=eg,f^c=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | #12167, #12168, #12169, … |
| maximal normal subgroup of | #12167, #12168, #12169, … |
| maximal quotient of | #10446, #10447, #10454, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 779);