Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
210 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)(6,7)(9,10)(11,14)(12,15)(13,16)(17,18,21,23)(19,24,20,22);(1,2,4,5)(3,6,8,7)(9,11)(10,12)(13,15)(14,16)(17,19,21,20)(18,22,23,24);(1,2)(3,7)(4,5)(6,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,24)(21,23);(1,3,4,8)(2,6,5,7)(9,12,13,14)(10,15,16,11)(17,20,21,19)(18,22,23,24)Polycyclic presentation:
2^7|a^2=fg,b^2=f,d^2=e,b^a=be,c^a=cf,c^b=cg,d^c=de32 conjugacy classes — view the full character table.
| Fitting subgroup of | #2644⋊C3 |
| maximal subgroup of | #29436, #29437, #29442, … |
| maximal normal subgroup of | #29436, #29437, #29442, … |
| maximal quotient of | #14260, #14266, #14331, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1376);