Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
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 | |
232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6,9,8)(7,10,12,11)(15,19)(17,20);(6,8)(10,11)(13,14,16,18)(15,17,19,20);(5,7,9,12)(6,10,8,11)(13,15,16,19)(14,17,18,20);(5,8,9,6)(7,11,12,10)Polycyclic presentation:
2^7|a^2=g,b^2=f,c^2=ef,d^2=e,b^a=be,c^a=cf,d^b=de50 conjugacy classes — view the full character table.
| maximal subgroup of | #8768, #8769, #8774, … |
| maximal normal subgroup of | #8768, #8769, #8774, … |
| maximal quotient of | #13804, #13806, #13810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1088);