Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| 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, prime radical, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(6,7)(9,10)(11,14)(12,15)(13,16)(17,18,21,19)(20,23,24,22);(1,2,5,3)(4,6,8,7)(9,11)(10,12)(13,15)(14,16)(17,19,21,18)(20,22,24,23);(1,3,5,2)(4,6,8,7)(9,11)(10,14)(12,16)(13,15)(17,20,21,24)(18,22,19,23);(1,4,5,8)(2,6,3,7)(9,12,13,14)(10,15,16,11)Polycyclic presentation:
2^7|a^2=g,b^2=f,c^2=f,d^2=e,b^a=be,c^a=cf,c^b=cg,d^c=de32 conjugacy classes — view the full character table.
| maximal subgroup of | #29133, #29134, #29135, … |
| maximal normal subgroup of | #29133, #29134, #29135, … |
| maximal quotient of | #14266, #14276, #14277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1379);