Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) = #233610 (order 512) · Inn(G) = C42 · Out(G) = #120 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C42 |
| commutator | C4 | C4×C8 |
| Frattini, cosocle | C4×C8 | C22 |
| socle, prime radical | C22 | #97 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(9,10,11,15,12,17,19,23,13,16,18,24,20,14,21,22);(1,2,3,5,4,6,7,8)(10,14,16,17)(11,18)(15,23,24,22)(19,21)Polycyclic presentation:
2^7|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,b^a=be,d^a=dg,c^b=cg44 conjugacy classes — view the full character table.
| maximal subgroup of | #7333, C2×#1313, #11404, … |
| maximal normal subgroup of | #7333, C2×#1313, #11404, … |
| maximal quotient of | #7053, #7137, #7333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 45);