Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) = C22×#511771 (order 2048) · Inn(G) = #120 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | #537 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,3)(5,6)(7,10)(8,11)(9,12);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12);(1,3,4,2)(6,10)(7,8)(11,12)Polycyclic presentation:
2^7|c^2=d,e^2=g,f^2=g,b^a=bd,c^a=ce,e^a=eg,c^b=cf,f^b=fg,e^c=eg,f^c=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7417, #12140, #12142, … |
| maximal normal subgroup of | #7417, #12140, #12142, … |
| maximal quotient of | #10402, #10405, #10406, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 771);