Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,20)(11,17,21,24)(12,16,22,23)(13,19,14,18);(2,5)(6,8)(9,11,14,22,15,21,13,12)(10,16,19,24,20,23,18,17);(1,3,4,7)(2,6,5,8)(9,12,13,21,15,22,14,11)(10,17,18,23,20,24,19,16);(11,21)(12,22)(16,23)(17,24)Polycyclic presentation:
2^7|a^2=g,b^2=f,c^2=e,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,d^b=dg,d^c=dg32 conjugacy classes — view the full character table.
| Frattini subgroup of | #94774 |
| maximal subgroup of | #9422, #9426, #9436, … |
| maximal normal subgroup of | #9422, #9426, #9436, … |
| maximal quotient of | #20764, #21094, #21202, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1871);