Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| 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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
212 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 212 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)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(2,6)(5,7)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)Polycyclic presentation:
2^7|d^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,c^b=ce,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #12104, #12107, #12280, … |
| maximal normal subgroup of | #12104, #12107, #12280, … |
| maximal quotient of | #21760, #21939, #22042, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1923);