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) = C2×C4 |
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 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ 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)(3,13,15,27)(4,10)(5,20)(6,22)(7,12)(8,14,23,28)(9,18)(11,21)(16,31,29,25)(17,32,30,24)(19,26);(1,3,5,16,21,30,19,28,7,15,18,29,6,17,4,14)(2,8,10,24,22,31,9,13,12,23,26,32,11,25,20,27)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=f,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 | #7422, #7423, C2×#1384, … |
| maximal normal subgroup of | #7422, #7423, C2×#1384, … |
| maximal quotient of | #7157, #7174, #7218, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 116);