Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 ◁ C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
273 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 273 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2)(3,4)(6,9)(7,12)(10,11);(5,6)(7,11)(8,10)(9,12);(2,4)(6,10)(9,11);(6,11)(7,12)(9,10)Polycyclic presentation:
2^7|e^2=g,b^a=be,c^a=cf,e^a=eg,c^b=cg,d^b=de,e^b=eg,e^d=eg29 conjugacy classes — view the full character table.
| maximal subgroup of | #19755, #19758, #19803, … |
| maximal normal subgroup of | #19755, #19758, #19803, … |
| maximal quotient of | #19883, #19890, #19898, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2004);