Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(6,10)(8,16)(11,13)(14,20);(5,6,9,10)(7,13,15,11)(8,12,16,17)(14,19,20,18);(1,2)(3,4);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19);(5,8)(6,12)(7,14)(9,16)(10,17)(11,18)(13,19)(15,20)Polycyclic presentation:
2^7|b^2=f,d^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| Frattini subgroup of | #94445 |
| Fitting subgroup of | #3486⋊C3, #11711508, #11713187, … |
| maximal subgroup of | #8455, #12710, #12712, … |
| maximal normal subgroup of | #8455, #12710, #12712, … |
| maximal quotient of | #34173, #34200, #34249, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2218);