Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 16384 |
| 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 | |
214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10)(11,13)(12,15)(14,16)(17,18,21,20)(19,23,24,22);(2,5)(6,8)(10,11)(14,15)(19,24)(22,23);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,19,21,24)(18,22,20,23);(1,3,4,7)(2,6,5,8)(9,12)(10,14)(11,15)(13,16)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^7|a^2=ef,c^2=f,d^2=eg,b^a=be,c^a=cf,d^a=de,c^b=cg38 conjugacy classes — view the full character table.
| maximal subgroup of | #10388, #10389, C2×#2608, … |
| maximal normal subgroup of | #10388, #10389, C2×#2608, … |
| maximal quotient of | #13953, #14092, #14094, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1340);