Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | Aut(G) = #72231730 (order 2048) · Inn(G) = #106 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #106 |
| commutator | C8 | C42 |
| Frattini, cosocle | C22×C8 | C22 |
| socle | C2 | #340 |
| prime radical | C23 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,11,8,13,7,23)(3,14,16,29,18,30,21,25)(4,12,6,22,20,28,19,10)(9,17,24,15,27,31,26,32);(1,3,6,17,5,21,20,32,8,18,19,31,7,16,4,15)(2,9,12,25,23,24,10,14,13,27,28,29,11,26,22,30)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=f,d^2=g,e^2=fg,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 | #7424, #7425, C2×#1385, … |
| maximal normal subgroup of | #7424, #7425, C2×#1385, … |
| maximal quotient of | #7157, #7173, #7220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 117);