Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 1024 |
| 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 | #496 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
184 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 184 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(7,8)(9,10)(11,18)(12,16)(13,21)(14,17)(15,20)(19,24)(22,23);(1,2,5,6)(3,7,4,8)(11,19)(15,23)(18,24)(20,22);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,20)(13,22)(14,19)(16,24)(17,23)(18,21);(1,4,5,3)(2,7,6,8)(9,12,14,13)(10,16,17,21)(11,20,19,22)(15,24,23,18)Polycyclic presentation:
2^7|b^2=e,c^2=e,d^2=eg,f^2=g,b^a=be,c^a=cf,f^a=fg,c^b=cg,d^b=de,f^c=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#3401, #61401, #61403, … |
| maximal normal subgroup of | C2×#3401, #61401, #61403, … |
| maximal quotient of | #22548, #22549, #22730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2133);