Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 16384 |
| 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 | |
780 conjugacy classes of subgroups; only the 428 normal subgroups are shown.
This group has 780 conjugacy classes of subgroups — too many to draw. Only its 428 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,13)(8,16)(9,10)(11,15)(12,17)(14,18)(19,20);(6,10)(11,13)(12,16)(18,19);(1,2)(3,4)(5,7,9,15)(6,11,10,13)(8,14,17,20)(12,18,16,19);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20);(5,8)(6,12)(7,14)(9,17)(10,16)(11,18)(13,19)(15,20)Polycyclic presentation:
2^7|c^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| Frattini subgroup of | #94446 |
| maximal subgroup of | #8462, #8539, #12706, … |
| maximal normal subgroup of | #8462, #8539, #12706, … |
| maximal quotient of | #34172, #34199, #34201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2217);