Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×D4×#392 (order 1024) · Inn(G) = D4 · Out(G) = C2×#392 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | D4 |
| commutator | C4 | C4×C8 |
| Frattini, cosocle | C2×C42 | C22 |
| socle | C22 | #95 |
| prime radical | C23 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2)(3,5)(4,7)(6,8)(9,10,12,15,14,16,13,11);(2,5,7,6)(9,11,13,16,14,15,12,10)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=f,d^2=g,e^2=g,b^a=bc,c^a=cf,e^a=ef56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1278, #7882, #7883, … |
| maximal normal subgroup of | C2×#1278, #7882, #7883, … |
| maximal quotient of | #7015, #7021, #7023, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 10);