Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) = #72249636 (order 8192) · Inn(G) = C2×C22⋊C4 · Out(G) = #23900 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C24 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,16,13,15,19,22)(11,14,17,21,18,20,23,24);(2,5)(6,8)(9,11,13,18)(10,14,15,20)(12,17,19,23)(16,21,22,24);(1,3,4,7)(2,6,5,8)(10,15)(12,19)(14,20)(17,23)Polycyclic presentation:
2^7|a^2=f,b^2=g,c^2=d,f^2=g,b^a=bd,c^a=ce,e^a=eg,f^c=fg32 conjugacy classes — view the full character table.
| Frattini subgroup of | #94528, #94529, #94531, … |
| maximal subgroup of | #7216, #7350, #7351, … |
| maximal normal subgroup of | #7216, #7350, #7351, … |
| maximal quotient of | #8699, #8700, #8702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 554);