Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | Aut(G) = #72236721 (order 2048) · Inn(G) = #386 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #386 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
156 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)(3,11)(4,15,17,8)(5,18)(6,10)(7,13)(9,20)(12,25,28,32)(14,26)(16,23,19,24)(21,27,31,29)(22,30);(1,3,5,13,6,14,20,30)(2,7,9,11,10,22,18,26)(4,12,19,29,17,28,16,27)(8,21,24,25,15,31,23,32);(1,4)(2,8)(3,12)(5,19)(6,17)(7,21)(9,24)(10,15)(11,25)(13,29)(14,28)(16,20)(18,23)(22,31)(26,32)(27,30)Polycyclic presentation:
2^7|a^2=f,b^2=df,d^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,e^b=eg,d^c=dg,f^c=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | #12756, #12757, #12779, … |
| maximal normal subgroup of | #12756, #12757, #12779, … |
| maximal quotient of | #9099, #9100, #9103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 641);