Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = #72231481 (order 2048) · Inn(G) = C22⋊C4 · Out(G) = #2021 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22⋊C4 |
| commutator | C22 | C4×C8 |
| Frattini, cosocle | C23×C4 | C22 |
| prime radical | C24 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 20 points:
(5,6)(7,11,12,9)(8,10)(13,14,15,17,16,18,19,20);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)Polycyclic presentation:
2^7|a^2=d,b^2=e,d^2=g,b^a=bc,c^a=cf,d^b=df44 conjugacy classes — view the full character table.
| maximal subgroup of | #7103, #7104, #7122, … |
| maximal normal subgroup of | #7103, #7104, #7122, … |
| maximal quotient of | #7016, #7017, #7020, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 12);