Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C25 |
| prime radical | #532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(5,8)(9,10)(11,18)(12,13)(14,15)(16,21)(17,22)(19,24)(20,23);(1,2,4,6)(3,5,7,8)(10,15)(12,17)(16,18)(19,23);(1,3,4,7)(2,5,6,8)(9,11,14,21)(10,16,15,18)(12,19,17,23)(13,20,22,24);(9,12,14,17)(10,13,15,22)(11,19,21,23)(16,20,18,24);(9,13)(10,17)(11,20)(12,15)(14,22)(16,23)(18,19)(21,24)Polycyclic presentation:
2^7|b^2=fg,c^2=g,d^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| maximal subgroup of | #8357, #8431, #8435, … |
| maximal normal subgroup of | #8357, #8431, #8435, … |
| maximal quotient of | #34187, #34206, #34213, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2239);