Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 32768 |
| 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 | |
258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,8,7)(10,14)(11,17)(12,20)(22,24);(5,7,8,6)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23);(1,2,3,4)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24);(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)Polycyclic presentation:
2^7|a^2=g,b^2=g,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=ce44 conjugacy classes — view the full character table.
| Frattini subgroup of | #72292967, #72292973, #72292986, … |
| maximal subgroup of | #9100, #9104, #9296, … |
| maximal normal subgroup of | #9100, #9104, #9296, … |
| maximal quotient of | #13779, #13786, #13790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1059);