Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| 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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
201 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 201 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,7)(4,6)(5,8)(9,10,13,18)(11,20,22,15)(12,19,14,17)(16,23,24,21);(1,3)(2,5)(4,8)(6,7)(9,11,13,22)(10,15,18,20)(12,23,14,21)(16,17,24,19);(3,8)(5,7)(9,11)(10,16)(12,21)(13,22)(14,23)(15,19)(17,20)(18,24);(2,6)(5,7)(9,12,13,14)(10,17,18,19)(11,21,22,23)(15,16,20,24)Polycyclic presentation:
2^7|a^2=g,b^2=g,d^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,c^b=ce,d^b=dg,f^c=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | #12119, #12303, C2×#3204, … |
| maximal normal subgroup of | #12119, #12303, C2×#3204, … |
| maximal quotient of | #21772, #21946, #22053, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1936);