Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) = #72234537 (order 8192) · Inn(G) = C24 · Out(G) = #6342371 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C24 |
| commutator | C22 | C23×C4 |
| prime radical | C2×#120 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
464 conjugacy classes of subgroups; only the 164 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 164 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2)(3,4)(6,8)(9,10,12,11);(2,4)(9,11,12,10);(5,6)(7,8);(1,2)(3,4)Polycyclic presentation:
2^7|a^2=g,b^2=g,b^a=be,c^a=cf,d^b=de50 conjugacy classes — view the full character table.
| maximal subgroup of | #12958, #13041, C2×#2358, … |
| maximal normal subgroup of | #12958, #13041, C2×#2358, … |
| maximal quotient of | C22⋊C42, #13813, #13814, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1090);