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, prime radical | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10,14,17)(11,15,13,21)(12,19,20,23)(16,22,24,18);(1,3,5,8)(2,6,4,7)(9,11,14,13)(10,15,17,21)(12,18,20,22)(16,23,24,19);(9,12,14,20)(10,16,17,24)(11,18,13,22)(15,23,21,19);(1,4,5,2)(3,6,8,7)Polycyclic presentation:
2^7|a^2=eg,b^2=eg,c^2=g,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 | #12353, #12357, #12405, … |
| maximal normal subgroup of | #12353, #12357, #12405, … |
| maximal quotient of | #22521, C4⋊C4×Q16, #22685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2115);