Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) = #72234537 (order 8192) · Inn(G) = C24 · Out(G) = #6342371 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | #392 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
284 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,10)(8,9)(11,12)(14,15)(18,19);(2,4)(6,8)(7,11)(13,14,17,15)(16,19,20,18);(1,2)(3,4)(5,7,9,11)(6,10,8,12)(13,15,17,14)(16,19,20,18);(5,8,9,6)(7,12,11,10)(13,16,17,20)(14,18,15,19)Polycyclic presentation:
2^7|b^2=f,c^2=e,d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de32 conjugacy classes — view the full character table.
| maximal subgroup of | #12149, #12243, #36709, … |
| maximal normal subgroup of | #12149, #12243, #36709, … |
| maximal quotient of | #14359, #14371, #14373, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1425);