Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
200 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,17)(11,14,18,22)(12,15,20,16)(19,23,24,21);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,21)(17,22)(20,24);(3,8)(5,7)(10,15)(11,18)(12,20)(14,21)(16,17)(22,23);(2,6)(5,7)Polycyclic presentation:
2^7|a^2=g,b^2=e,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,c^b=ceg,f^c=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | #9334, #9379, #10008, … |
| maximal normal subgroup of | #9334, #9379, #10008, … |
| maximal quotient of | #21808, #21866, #21986, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1947);