Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = #72231803 (order 2048) · Inn(G) = D4 · Out(G) = C2×#2021 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | D4 |
| commutator | C4 | C4×C8 |
| Frattini, cosocle | C2×C42 | C22 |
| socle, prime radical | C23 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2)(3,4)(6,10)(7,8)(11,12)(13,14,15,17,16,18,19,20);(2,4)(5,6,8,11,9,12,7,10)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=f,d^2=g,e^2=f,b^a=bc,c^a=cf,e^a=ef56 conjugacy classes — view the full character table.
| maximal subgroup of | #7112, #7113, #7120, … |
| maximal normal subgroup of | #7112, #7113, #7120, … |
| maximal quotient of | #7015, #7018, #7020, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 9);