Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = #10586342 (order 512) · Inn(G) = C22⋊C4 · Out(G) = C25 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22⋊C4 |
| commutator | C22 | C4×C8 |
| Frattini, cosocle | C22×C8 | C22 |
| socle | C2 | #336 |
| prime radical | C23 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,8,5,10,15,22,6,9,16,21,17,23,28,31)(3,11,13,24,14,25,27,32,12,7,18,19,26,20,29,30);(1,3)(2,7,9,11)(4,13)(5,14)(6,12)(8,19,21,24)(10,20,23,25)(15,27)(16,18)(17,26)(22,30,31,32)(28,29)Polycyclic presentation:
2^7|a^2=d,b^2=e,d^2=f,f^2=g,b^a=bc,e^a=eg,c^b=cg44 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1378, #12597, #12609, … |
| maximal normal subgroup of | C2×#1378, #12597, #12609, … |
| maximal quotient of | #7140, #7144, #7147, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 110);