Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,8,10)(7,11,12,9)(13,14,15,18,16,19,20,17);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,17)(15,20)(18,19);(2,4)(14,17)(15,20)(18,19)Polycyclic presentation:
2^7|a^2=d,b^2=deg,d^2=g,e^2=g,b^a=bd,c^a=ce,c^b=cf,d^b=dg,e^b=eg,d^c=dg,e^c=eg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7434, #7786, #7789, … |
| maximal normal subgroup of | #7434, #7786, #7789, … |
| maximal quotient of | #10660, #10667, #10671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 825);