Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | Aut(G) = #72231836 (order 2048) · Inn(G) = #95 · Out(G) = #392 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #95 |
| commutator | C23 | C42 |
| Frattini, cosocle | C23×C4 | C22 |
| prime radical | C24 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17,18,14)(12,16)(13,15)(19,23,24,21)(20,22);(1,3)(2,5,6,7)(4,8)(9,11,12,19,13,18,20,24)(10,14,16,21,15,17,22,23)Polycyclic presentation:
2^7|a^2=d,b^2=e,e^2=f,b^a=bc,c^a=cf,e^a=eg,c^b=cg,d^b=df26 conjugacy classes — view the full character table.
| maximal subgroup of | #7395, #7427, C2×#1306, … |
| maximal normal subgroup of | #7395, #7427, C2×#1306, … |
| maximal quotient of | #7032, #7033, #7037, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 38);