Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) = C24×#120 (order 512) · Inn(G) = #120 · Out(G) = C24 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
160 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 160 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) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,7)(3,4)(8,15)(9,13)(10,16)(12,14);(1,2)(3,7)(4,13)(5,9)(6,10)(8,11)(12,16)(14,15);(1,3,11,14)(2,7,8,15)(4,6,12,5)(9,13,10,16)Polycyclic presentation:
2^7|c^2=fg,d^2=f,b^a=bd,c^a=ce,d^a=df,d^b=df,e^b=eg,d^c=dg23 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1600, #18114, #18126, … |
| maximal normal subgroup of | C2×#1600, #18114, #18126, … |
| maximal quotient of | #7737, #7740, #7742, … |
| outer automorphism group of | #28250, #29231 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 332);