Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×C4×AGL(1,ℤ/8) (order 256) · Inn(G) = D4 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | D4 |
| commutator | C4 | C22×C8 |
| Frattini, cosocle | C2×C8 | C23 |
| socle | C2 | C8×D4 |
| prime radical | #135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
73 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10,5,11,20,28,7,12,22,29,21,9,25,27)(3,13,15,18,16,24,30,19,17,26,31,32,14,8,23,4);(1,3)(2,8)(4,10)(5,16)(6,15)(7,17)(9,26)(11,13)(12,24)(14,21)(18,28)(19,29)(20,30)(22,31)(23,25)(27,32);(1,4,6,13,5,18,20,24,7,19,22,26,21,32,25,8)(2,3,10,15,11,16,28,30,12,17,29,31,9,14,27,23)Polycyclic presentation:
2^7|a^2=e,c^2=e,d^2=g,e^2=f,f^2=g,b^a=bd,d^a=dg,d^b=dg56 conjugacy classes — view the full character table.
| maximal subgroup of | #7381, #7382, #12046, … |
| maximal normal subgroup of | #7381, #7382, #12046, … |
| maximal quotient of | C16×D8, #12039, C16×SD16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 902);