Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | Aut(G) = #72231699 (order 1024) · Inn(G) = #102 · Out(G) = C25 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #102 |
| commutator | C8 | C42 |
| Frattini, cosocle | C22×C8 | C22 |
| socle | C2 | #340 |
| prime radical | C23 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,10,7,11,6,22)(3,12,14,29,17,25,16,31)(4,18,19,27,21,32,20,9)(8,23,24,15,26,30,28,13);(1,3,4,15,5,14,19,30,7,17,21,13,6,16,20,23)(2,8,9,25,22,28,32,29,11,26,27,12,10,24,18,31)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=fg,d^2=g,e^2=fg,f^2=g,b^a=bc,c^a=cf,e^a=ef,f^a=fg,c^b=cg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1389, #12646, #12649, … |
| maximal normal subgroup of | C2×#1389, #12646, #12649, … |
| maximal quotient of | #7159, #7175, #7177, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 121);