Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 2048 |
| 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 | |
521 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24);(10,14)(15,17)(16,20)(22,23);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)Polycyclic presentation:
2^7|c^2=f,e^2=fg,b^a=bf,c^a=cg,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| maximal subgroup of | #61486, #61487, #61500, … |
| maximal normal subgroup of | #61486, #61487, #61500, … |
| maximal quotient of | #34195, #34234, #34237, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2253);