Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 147456 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
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 | |
216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,13)(14,16)(18,19)(22,23);(2,3)(4,8)(10,13)(11,16)(17,18,21,19)(20,23,24,22);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,16)(13,15)(17,19,21,18)(20,23,24,22);(1,4,5,8)(2,6,3,7)(9,10)(11,14)(12,13)(15,16)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^7|a^2=ef,b^2=eg,c^2=fg,d^2=ef,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de32 conjugacy classes — view the full character table.
| derived subgroup of | #73290, #73291, #7749⋊C3, … |
| Fitting subgroup of | #73203, #73290, #73291, … |
| maximal subgroup of | #7749, #7759, #7812, … |
| maximal normal subgroup of | #7749, #7759, #7812, … |
| maximal quotient of | #14467, #14601, #15510, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1543);