Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) = C2×#72237160 (order 2048) · Inn(G) = C2×D8 · Out(G) = C22×#41 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| prime radical | #493 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
87 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(6,10)(7,13)(8,15)(11,20)(12,19)(14,17)(16,18);(1,2,3,4);(5,6,9,12)(7,14,16,20)(8,17,15,11)(10,18,19,13)Polycyclic presentation:
2^7|b^2=d,c^2=g,e^2=fg,f^2=g,b^a=bd,c^a=ce,e^a=ef,f^a=fg,e^c=ef,f^c=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | #12317, #12318, #12325, … |
| maximal normal subgroup of | #12317, #12318, #12325, … |
| maximal quotient of | #12417, #12425, #12446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 939);