Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) = #72263067 (order 8192) · Inn(G) = C2×D4 · Out(G) = #449475 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C22 | C2×C4⋊C4 |
| prime radical | C23 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,19,15,20,14,18)(11,17,12,16,22,23,21,24);(2,5)(6,8)(9,11,14,21,15,22,13,12)(10,16,19,23,20,24,18,17);(1,3,4,7)(2,6,5,8)(9,12,13,22,15,21,14,11)(10,17,18,24,20,23,19,16)Polycyclic presentation:
2^7|a^2=f,b^2=e,c^2=d,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg44 conjugacy classes — view the full character table.
| Frattini subgroup of | #94561, #94562, #94564, … |
| maximal subgroup of | #7219, #7220, #7223, … |
| maximal normal subgroup of | #7219, #7220, #7223, … |
| maximal quotient of | #8784, #8785, #8789, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 569);