Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×#2955 (order 256) · Inn(G) = D4 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | D4 |
| commutator | C4 | C2×C16 |
| Frattini, cosocle | C2×C16 | C22 |
| socle | C2 | C4⋊C16 |
| prime radical | C22 | M32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
29 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,11,6,12,20,27,7,10,21,25,17,26,30,32,8,13,22,28,15,9,23,24,4,18,19,31,3,14,16,29);(1,3,4,15,8,17,7,6)(2,9,10,14,13,12,18,26)(5,16,19,23,22,30,21,20)(11,24,25,29,28,27,31,32)Polycyclic presentation:
2^7|a^2=d,b^2=c,c^2=g,d^2=e,e^2=f,f^2=g,b^a=bc,c^a=cg56 conjugacy classes — view the full character table.
| maximal subgroup of | #7381, #7382, #7447, … |
| maximal normal subgroup of | #7381, #7382, #7447, … |
| maximal quotient of | #7455, #7458, #7467, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 154);