Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10,14,18)(11,16,20,24)(12,21,19,15)(13,23,22,17);(5,6)(7,8)(9,11,14,20)(10,15,18,21)(12,22,19,13)(16,17,24,23);(2,4)(6,8)(9,12,13,11,14,19,22,20)(10,16,17,21,18,24,23,15)Polycyclic presentation:
2^7|a^2=g,b^2=g,c^2=d,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7082, #7090, #12161, … |
| maximal normal subgroup of | #7082, #7090, #12161, … |
| maximal quotient of | #10478, #10479, #10482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 788);