Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) = #72237685 (order 2048) · Inn(G) = #121 · Out(G) = #379 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C23 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10,13,16,14,18,12,17)(11,15,19,24,21,22,20,23);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,20)(13,19)(14,21)(16,24)(17,23)(18,22);(1,4,5,3)(2,6,7,8)(10,16)(11,19)(12,13)(15,22)(17,18)(20,21)Polycyclic presentation:
2^7|a^2=eg,b^2=d,c^2=d,e^2=g,f^2=g,b^a=bd,c^a=ce,c^b=cf,f^b=fg,e^c=eg,f^c=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2090, #23674, #23675, … |
| maximal normal subgroup of | C2×#2090, #23674, #23675, … |
| maximal quotient of | #10653, #10654, #10655, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 822);