Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = #72234537 (order 8192) · Inn(G) = C23 · Out(G) = C2×#6342371 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C23 |
| commutator | C22 | C2×C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
306 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 128 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,4)(5,6,8,7)(10,12)(13,14,16,15);(9,10)(11,12)(13,14,16,15);(1,2)(3,4)(5,7,8,6)(9,10)(11,12)(13,15,16,14)Polycyclic presentation:
2^7|a^2=f,b^2=g,c^2=f,b^a=bd,c^a=ce56 conjugacy classes — view the full character table.
| maximal subgroup of | #7215, #8347, C2×#1436, … |
| maximal normal subgroup of | #7215, #8347, C2×#1436, … |
| maximal quotient of | #7554, #7556, #7557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 168);