Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | Aut(G) = #511771 (order 512) · Inn(G) = D42 · Out(G) = D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
232 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,7)(4,5)(8,15)(9,13)(10,16)(11,14);(1,2)(3,10)(4,13)(5,7)(6,9)(8,11)(12,15)(14,16);(1,2)(3,8)(4,7)(5,13)(6,9)(10,11)(12,16)(14,15);(1,3)(2,8)(4,14)(5,11)(6,12)(7,15)(9,16)(10,13)Polycyclic presentation:
2^7|e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,d^b=de,e^b=eg,f^c=fg,e^d=eg26 conjugacy classes — view the full character table.
| maximal subgroup of | #13677, #13679, C2×#3288, … |
| maximal normal subgroup of | #13677, #13679, C2×#3288, … |
| maximal quotient of | #19755, #19756, #19759, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2020);