Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(10,15)(11,12)(13,21)(23,24);(2,6)(3,7)(9,10,14,15)(11,17,12,16)(13,18,21,22)(19,24,20,23);(9,11,14,12)(10,16,15,17)(13,19,21,20)(18,23,22,24);(1,3,5,7)(2,4,6,8)(9,12,14,11)(10,17,15,16)(13,20,21,19)(18,24,22,23);(1,4)(2,7)(3,6)(5,8)(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)Polycyclic presentation:
2^7|b^2=g,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df38 conjugacy classes — view the full character table.
| maximal subgroup of | #8499, #18694, #18695, … |
| maximal normal subgroup of | #8499, #18694, #18695, … |
| maximal quotient of | #34371, #34384, #34417, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2284);