Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | #532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,20)(12,15)(13,17)(14,19)(16,23)(18,21)(22,24);(2,5)(6,8)(9,11,21,22,14,23,13,12)(10,15,17,16,19,24,18,20);(1,3,4,7)(2,6,5,8)(9,12,13,23,14,22,21,11)(10,16,18,15,19,20,17,24);(9,13,14,21)(10,17,19,18)(11,22,23,12)(15,20,24,16)Polycyclic presentation:
2^7|b^2=fg,c^2=e,d^2=g,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,d^b=dg,d^c=dg32 conjugacy classes — view the full character table.
| Frattini subgroup of | #94771, #94773 |
| maximal subgroup of | #9442, #10019, #10021, … |
| maximal normal subgroup of | #9442, #10019, #10021, … |
| maximal quotient of | #20765, #21093, #21201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1874);