Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | Aut(G) = C23×B(4,𝔽2) (order 512) · Inn(G) = #99 · Out(G) = C24 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #99 |
| commutator | C23 | C42 |
| Frattini, prime radical, cosocle | C22×D4 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
117 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 117 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 12 points:
(5,6)(7,11,12,9)(8,10);(1,2,3,4)(6,9)(10,11)Polycyclic presentation:
2^7|a^2=d,b^2=e,b^a=bc,c^a=cf,f^a=fg,d^b=df,d^c=dg26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1390, #12573, #12574, … |
| maximal normal subgroup of | C2×#1390, #12573, #12574, … |
| maximal quotient of | #7187, #7203, #7204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 122);