Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) = #72262293 (order 8192) · Inn(G) = C2×C22⋊C4 · Out(G) = D4×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C24 | C23 |
| prime radical | C23×D4 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2)(3,4)(5,6)(7,11,12,9)(8,10);(5,7)(6,9)(8,12)(10,11);(2,4)Polycyclic presentation:
2^7|a^2=f,b^a=bd,c^a=ce,d^a=dg,f^b=fg32 conjugacy classes — view the full character table.
| derived subgroup of | #72227523, #72227524, #72227525, … |
| Frattini subgroup of | #94514, #94515, #94516, … |
| maximal subgroup of | #7304, #7307, #7310, … |
| maximal normal subgroup of | #7304, #7307, #7310, … |
| maximal quotient of | #8298, #8299, #8301, … |
| outer automorphism group of | #1561 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 513);