Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C23 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
169 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,18)(11,15,20,28)(12,22,25,37)(13,26,23,17)(16,30,31,24)(19,32,34,40)(21,36,33,29)(27,38,39,35);(2,5)(6,8)(9,11,14,20)(10,15,18,28)(12,19,25,34)(13,21,23,33)(16,27,31,39)(17,29,26,36)(22,32,37,40)(24,35,30,38);(9,12,23,30,14,25,13,24)(10,16,26,37,18,31,17,22)(11,19,33,38,20,34,21,35)(15,27,36,40,28,39,29,32);(1,3,4,7)(2,6,5,8)(11,20)(15,28)(19,34)(21,33)(27,39)(29,36)(32,40)(35,38)Polycyclic presentation:
2^7|a^2=g,b^2=g,c^2=fg,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=dg32 conjugacy classes — view the full character table.
| maximal subgroup of | #11771, #11914, #11990, … |
| maximal normal subgroup of | #11771, #11914, #11990, … |
| maximal quotient of | #20821, #21281, #21289, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1909);