Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| 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 | #469 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
185 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 185 conjugacy classes of subgroups — too many to draw. Only its 88 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,19)(13,22)(14,17)(16,20)(18,24)(21,23);(2,5)(6,8)(9,11,14,13)(10,15,17,22)(12,20,21,18)(16,23,24,19);(9,12,13,20,14,21,11,18)(10,16,15,19,17,24,22,23);(1,3,4,7)(2,6,5,8)(12,21)(16,24)(18,20)(19,23)Polycyclic presentation:
2^7|b^2=g,c^2=f,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,c^b=cg,d^c=dg32 conjugacy classes — view the full character table.
| maximal subgroup of | #11836, #33690, C2×#3183, … |
| maximal normal subgroup of | #11836, #33690, C2×#3183, … |
| maximal quotient of | #20825, #21286, #21287, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1915);