Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| 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×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
193 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 193 conjugacy classes of subgroups — too many to draw. Only its 90 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)(9,10,14,18)(11,15,21,24)(12,17,13,16)(19,23,20,22);(2,5)(6,8)(9,11,14,21)(10,15,18,24)(12,19,13,20)(16,22,17,23);(10,16)(12,13)(15,22)(17,18)(19,20)(23,24);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,17,18,16)(11,19,21,20)(15,23,24,22)Polycyclic presentation:
2^7|a^2=g,b^2=g,d^2=eg,f^2=g,b^a=be,c^a=cf,f^a=fg,d^c=dg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #9159, #9160, #9312, … |
| maximal normal subgroup of | #9159, #9160, #9312, … |
| maximal quotient of | #20743, #21121, #21123, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1848);