Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | Aut(G) order 32768 |
| 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 | |
212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(6,9)(7,11)(14,18)(15,21)(20,23)(22,27);(6,9)(10,12)(13,14,17,18)(15,19,21,25)(16,23,24,20)(22,28,27,26);(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,28)(24,27);(5,7)(6,10)(8,11)(9,12)(13,16,17,24)(14,20,18,23)(15,22,21,27)(19,26,25,28)Polycyclic presentation:
2^7|a^2=g,b^2=e,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de38 conjugacy classes — view the full character table.
| maximal subgroup of | #9426, #9427, #9429, … |
| maximal normal subgroup of | #9426, #9427, #9429, … |
| maximal quotient of | #13830, #13833, #13834, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1113);