Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
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 | |
204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,5)(3,8,4,7)(9,10,13,16)(11,14,15,12)(19,23)(21,24);(1,3,6,4)(2,7,5,8)(9,11,13,15)(10,14,16,12)(17,18,20,22)(19,21,23,24);(1,4,6,3)(2,8,5,7)(9,12,13,14)(10,15,16,11);(1,5,6,2)(3,8,4,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,23)(22,24)Polycyclic presentation:
2^7|a^2=f,b^2=eg,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=dg44 conjugacy classes — view the full character table.
| Fitting subgroup of | #2863⋊C3 |
| maximal subgroup of | #7669, #7673, #7689, … |
| maximal normal subgroup of | #7669, #7673, #7689, … |
| maximal quotient of | #14688, #14699, #14703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1595);