Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | Aut(G) = #72231669 (order 1024) · Inn(G) = #106 · Out(G) = C25 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #106 |
| commutator | C8 | C42 |
| Frattini, cosocle | C22×C8 | C22 |
| prime radical | C23 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2)(3,4)(6,11,12,13)(7,15)(8,17)(9,16)(14,20,18,19);(2,4)(5,6,8,13,9,14,17,20,10,12,7,11,16,18,15,19)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=f,e^2=f,f^2=g,b^a=bc,c^a=cf,e^a=ef,f^a=fg,d^b=dg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1383, #12563, #12647, … |
| maximal normal subgroup of | C2×#1383, #12563, #12647, … |
| maximal quotient of | #7156, #7173, #7174, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 115);