Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 12288 |
| 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 | |
505 conjugacy classes of subgroups; only the 390 normal subgroups are shown.
This group has 505 conjugacy classes of subgroups — too many to draw. Only its 390 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,8)(9,10,15,13)(11,18,12,16)(14,21,22,17)(19,23,20,24);(9,11,15,12)(10,16,13,18)(14,19,22,20)(17,23,21,24);(1,2)(3,6)(4,5)(7,8)(9,12,15,11)(10,16,13,18)(14,20,22,19)(17,23,21,24);(9,13,15,10)(11,16,12,18)(14,21,22,17)(19,23,20,24);(1,3,4,7)(2,6,5,8)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)Polycyclic presentation:
2^7|a^2=f,b^2=f,c^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 | #3518⋊C3, #11711037, #11712028, … |
| maximal subgroup of | #8413, #17654, #17683, … |
| maximal normal subgroup of | #8413, #17654, #17683, … |
| maximal quotient of | #34193, #34231, #34242, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2250);