Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) = #72215858 (order 1024) · Inn(G) = C22⋊C4 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22⋊C4 |
| commutator | C2×C4 | C2×C8 |
| Frattini, cosocle | C2×C42 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
83 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(10,13,14,15)(17,18,19,21,20,22,23,24);(1,3)(2,5)(4,8)(6,7)(9,10,12,14)(11,15,16,13)Polycyclic presentation:
2^7|a^2=d,b^2=f,c^2=f,d^2=g,b^a=bc,c^a=ce,c^b=cf,d^b=def38 conjugacy classes — view the full character table.
| maximal subgroup of | #7345, #7376, #7409, … |
| maximal normal subgroup of | #7345, #7376, #7409, … |
| maximal quotient of | #7025, #7128, #7129, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 4);