Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C23 |
| commutator | C23 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,7,8)(10,12);(9,10)(11,12)(13,14,16,18)(15,19,20,17);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15,16,20)(14,17,18,19)Polycyclic presentation:
2^7|a^2=g,b^2=f,c^2=f,b^a=bd,c^a=ce,c^b=cf44 conjugacy classes — view the full character table.
| derived subgroup of | #72267642, #72267806, #72267850 |
| Frattini subgroup of | #93193, #93215, #93216, … |
| Fitting subgroup of | #1446⋊C3, #11710951, #11711140, … |
| maximal subgroup of | #7131, #7308, #8455, … |
| maximal normal subgroup of | #7131, #7308, #8455, … |
| maximal quotient of | #7557, #11380, #11381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 178);