Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,5,7,8)(10,12)(14,15)(17,18,21,24)(19,23,20,22);(1,2)(3,5)(4,6)(7,8)(9,10,13,12)(11,14,16,15)(17,19,21,20)(18,22,24,23);(1,2)(3,5)(4,6)(7,8)(9,11,13,16)(10,14,12,15)(17,18)(19,22)(20,23)(21,24);(1,3,4,7)(2,5,6,8)(9,12,13,10)(11,15,16,14)(17,20,21,19)(18,23,24,22)Polycyclic presentation:
2^7|a^2=f,b^2=g,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=dg44 conjugacy classes — view the full character table.
| maximal subgroup of | #7657, #7677, #7697, … |
| maximal normal subgroup of | #7657, #7677, #7697, … |
| maximal quotient of | #14689, #14699, #14704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1596);