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 | |
530 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)(3,7)(9,10)(11,17)(12,14)(13,20)(15,18)(16,21)(19,22)(23,24);(2,5)(6,8)(10,12)(11,18)(16,20)(19,24);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24);(1,3)(2,6)(4,7)(5,8)(9,12,14,10)(11,17,18,15)(13,20,21,16)(19,23,24,22);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)Polycyclic presentation:
2^7|c^2=fg,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 | #8528, #12768, #12781, … |
| maximal normal subgroup of | #8528, #12768, #12781, … |
| maximal quotient of | #34366, #34390, #34408, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2277);