Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
187 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 187 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,4)(5,6)(7,8)(9,10,13,16,14,11,12,15);(1,2,3,4)(9,11,14,10)(12,15,13,16);(6,8)(9,10,14,11)(12,16,13,15)Polycyclic presentation:
2^7|a^2=f,b^2=dfg,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=eg,f^c=fg38 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1880, #20924, #20926, … |
| maximal normal subgroup of | C2×#1880, #20924, #20926, … |
| maximal quotient of | #8948, #8952, #8978, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 612);