Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) = #72232371 (order 2048) · Inn(G) = C2×D4 · Out(G) = #3484 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C42 | C23 |
| socle | C22 | C2×C22⋊C4 |
| prime radical | C23 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,20,24,21,23,27,30)(19,22,26,28,25,29,31,32);(1,3,6,13)(2,7,9,15)(4,12,5,11)(8,16,14,10)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(18,23)(19,25)(24,30)(26,31)Polycyclic presentation:
2^7|a^2=e,b^2=f,c^2=f,d^2=f,e^2=g,b^a=bd,c^a=cg,d^a=df,c^b=cfg,d^b=df32 conjugacy classes — view the full character table.
| maximal subgroup of | #8490, #8501, #8502, … |
| maximal normal subgroup of | #8490, #8501, #8502, … |
| maximal quotient of | #7701, #7702, #7703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 229);