Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | Aut(G) = #499261 (order 512) · Inn(G) = #106 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #106 |
| commutator | C8 | C2×C8 |
| Frattini, cosocle | C4×C8 | C22 |
| socle | C2 | #334 |
| prime radical | C22 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,11,7,12,22,28,8,13,21,10,6,24,4,20)(3,14,16,25,18,29,15,26,19,27,23,32,17,31,30,9);(1,3,4,15,6,17,21,16,8,19,22,30,7,18,5,23)(2,9,10,14,12,25,20,29,13,26,11,27,24,32,28,31)Polycyclic presentation:
2^7|a^2=d,b^2=c,c^2=e,d^2=f,e^2=g,f^2=g,b^a=bc,c^a=ce,e^a=eg,d^b=dg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7390, #7391, #7457, … |
| maximal normal subgroup of | #7390, #7391, #7457, … |
| maximal quotient of | #7099, #7161, C2×#1369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 101);