Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | Aut(G) order 28672 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C24 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
209 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(10,13)(11,15)(18,20)(22,23);(9,10)(11,14)(12,13)(15,16)(17,18,21,20)(19,22,24,23);(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,24)(18,22,20,23);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^7|b^2=fg,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=dg44 conjugacy classes — view the full character table.
| Fitting subgroup of | #2865⋊C7 |
| maximal subgroup of | #35027, #35028, #35040, … |
| maximal normal subgroup of | #35027, #35028, #35040, … |
| maximal quotient of | #14690, #14701, #14704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1597);