Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) = #72232993 (order 2048) · Inn(G) = C2×C22⋊C4 · Out(G) = C23×D4 |
| 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 ▷ C22 ▷ C1 | |
135 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)(3,7,8,5)(4,6)(9,10,13,17,14,18,12,16)(11,15,19,22,20,23,21,24);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,21)(13,19)(14,20)(16,24)(17,22)(18,23);(9,12,14,13)(10,16,18,17)(11,19,20,21)(15,22,23,24)Polycyclic presentation:
2^7|a^2=e,b^2=f,c^2=g,e^2=g,b^a=bd,d^a=df,c^b=cg,e^b=ef32 conjugacy classes — view the full character table.
| maximal subgroup of | #8467, #8468, #8537, … |
| maximal normal subgroup of | #8467, #8468, #8537, … |
| maximal quotient of | #7665, #7668, #7671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 204);