Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | Aut(G) = #72252587 (order 4096) · Inn(G) = C24 · Out(G) = #33551 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C24 |
| commutator, socle | C22 | C23×C4 |
| prime radical | #532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
197 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)(3,7)(4,6)(5,8)(9,10,12,16,13,17,20,24)(11,14,19,21,18,15,22,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,22)(23,24);(3,8)(5,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24);(2,6)(5,7)(11,18)(14,15)(19,22)(21,23)Polycyclic presentation:
2^7|a^2=f,f^2=g,b^a=be,c^a=cg,d^a=de,c^b=ce,d^b=dg,d^c=dg38 conjugacy classes — view the full character table.
| maximal subgroup of | #8918, #9005, #9007, … |
| maximal normal subgroup of | #8918, #9005, #9007, … |
| maximal quotient of | #20532, #20539, #20563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1712);