Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C23 | C24 |
| commutator | C22 | C23×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,4)(5,6,7,8)(9,10,14,13)(11,16,12,15);(9,11,14,12)(10,15,13,16);(1,2,3,4)(9,12,14,11)(10,15,13,16);(9,13,14,10)(11,15,12,16)Polycyclic presentation:
2^7|a^2=g,b^2=e,c^2=ef,d^2=e,b^a=be,c^a=cf,d^b=de50 conjugacy classes — view the full character table.
| Fitting subgroup of | #2351⋊C3, #11711174, #11711175, … |
| maximal subgroup of | #8766, #8767, #9176, … |
| maximal normal subgroup of | #8766, #8767, #9176, … |
| maximal quotient of | #13804, Q8×#95, #13821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1083);