Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = #72218016 (order 2048) · Inn(G) = C22 · Out(G) = #234591 |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C4×C8 | C22 |
| commutator | C2 | C82 |
| socle, prime radical | C22 | C4×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 24 points:
(9,10,11,15,12,16,17,22,13,14,18,20,19,21,23,24);(1,2,3,5,4,6,7,8)(10,14)(15,20)(16,21)(22,24)Polycyclic presentation:
2^7|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,b^a=bg80 conjugacy classes — view the full character table.
| Frattini subgroup of | #93668, #93757, #93758, … |
| maximal subgroup of | #7053, #7074, #7075, … |
| maximal normal subgroup of | #7053, #7074, #7075, … |
| maximal quotient of | #7052, #7136, #7330, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 43);