Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
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 | |
222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(14,16)(15,19);(1,3,4,2)(6,9)(10,11)(13,14,17,16)(15,20,19,18);(5,6)(7,10)(8,9)(11,12)(13,15,17,19)(14,18,16,20);(5,7,8,12)(6,10,9,11)(13,16,17,14)(15,20,19,18)Polycyclic presentation:
2^7|a^2=g,b^2=fg,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=ce44 conjugacy classes — view the full character table.
| maximal subgroup of | #21619, #21620, #21621, … |
| maximal normal subgroup of | #21619, #21620, #21621, … |
| maximal quotient of | #13779, #13783, #13792, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1062);