Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) order 4096 |
| 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 | C2×D8 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
180 conjugacy classes of subgroups.
| 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)(11,14)(12,15)(13,17)(16,20)(18,21)(19,23)(22,24);(1,2,5,6)(3,7,4,8)(9,11,13,19)(10,14,17,23)(12,18,20,24)(15,21,16,22);(1,3,5,4)(2,7,6,8)(10,15)(12,20)(14,21)(16,17)(18,24)(22,23);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^7|b^2=eg,c^2=e,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=de,f^c=fg35 conjugacy classes — view the full character table.
| maximal subgroup of | #12331, #12335, #12383, … |
| maximal normal subgroup of | #12331, #12335, #12383, … |
| maximal quotient of | #22541, #22727, #22807, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2125);