Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C2×D4 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
194 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18);(1,3)(2,5)(4,8)(6,7);(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17);(3,8)(5,7)Polycyclic presentation:
2^7|a^2=eg,c^2=g,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=de,f^c=fg35 conjugacy classes — view the full character table.
| maximal subgroup of | #19781, #19782, #19797, … |
| maximal normal subgroup of | #19781, #19782, #19797, … |
| maximal quotient of | D4×#103, #22487, #22575, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2031);