Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| 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 | #525 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
207 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 207 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)(11,15)(12,19)(13,22)(14,17)(16,18)(20,24)(21,23);(1,3)(2,5)(4,8)(6,7)(9,11,14,13)(10,15,17,22)(12,20,21,18)(16,19,24,23);(3,8)(5,7)(9,12,14,21)(10,16,17,24)(11,18,13,20)(15,23,22,19);(2,6)(5,7)Polycyclic presentation:
2^7|b^2=g,c^2=g,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 | #9331, #18323, #18331, … |
| maximal normal subgroup of | #9331, #18323, #18331, … |
| maximal quotient of | #21768, #21769, #21944, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1928);