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) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | #550 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
199 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(1,2,4,6)(3,7,8,5)(9,10,14,18)(11,19,20,24)(12,17,13,16)(15,22,23,21);(1,3)(2,5)(4,8)(6,7)(11,20)(15,23)(19,24)(21,22);(9,11,14,20)(10,15,18,23)(12,21,13,22)(16,19,17,24);(3,8)(5,7)(9,12,14,13)(10,16,18,17)(11,21,20,22)(15,19,23,24)Polycyclic presentation:
2^7|a^2=eg,c^2=g,d^2=g,f^2=g,b^a=be,c^a=cf,d^a=dg,f^a=fg,c^b=cg,d^b=de,f^c=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | #19781, #19785, #19829, … |
| maximal normal subgroup of | #19781, #19785, #19829, … |
| maximal quotient of | #22432, #22502, #22579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2065);