Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 6144 |
| 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 | |
506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,8)(10,13)(11,17)(16,20)(18,23);(9,10,14,13)(11,15,17,19)(12,20,21,16)(18,24,23,22);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23);(9,12,14,21)(10,16,13,20)(11,18,17,23)(15,22,19,24);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,19,17,15)(12,20,21,16)(18,24,23,22)Polycyclic presentation:
2^7|b^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| Fitting subgroup of | #3516⋊C3, #11715192, #11715193, … |
| maximal subgroup of | #61472, #61473, #61490, … |
| maximal normal subgroup of | #61472, #61473, #61490, … |
| maximal quotient of | #34192, #34230, #34241, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2248);