Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) = #72262696 (order 4096) · Inn(G) = C2×C22⋊C4 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator, prime radical | C23 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
136 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)(5,6,8,12,9,11,17,19)(7,13,14,20,15,10,16,18);(1,2,3,4);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)Polycyclic presentation:
2^7|a^2=f,b^2=d,c^2=g,f^2=g,b^a=bd,c^a=ce,e^a=eg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7238, #7323, #8354, … |
| maximal normal subgroup of | #7238, #7323, #8354, … |
| maximal quotient of | #8558, #8560, #8562, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 533);