Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 49152 |
| 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 | C23 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,14,18)(11,16,12,15)(13,17,21,24)(19,23,20,22);(9,11,14,12)(10,15,18,16)(13,20,21,19)(17,23,24,22);(1,3,4,8)(2,5,6,7)(9,12,14,11)(10,16,18,15)(13,20,21,19)(17,23,24,22);(9,13,14,21)(10,17,18,24)(11,19,12,20)(15,22,16,23);(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)Polycyclic presentation:
2^7|a^2=g,b^2=f,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.
| Fitting subgroup of | #3502⋊C3, #11711046, #11712032, … |
| maximal subgroup of | #8356, #8433, #8460, … |
| maximal normal subgroup of | #8356, #8433, #8460, … |
| maximal quotient of | #34182, #34213, #34249, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2234);