Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | Aut(G) = C22×B(4,𝔽2) (order 256) · Inn(G) = B(4,𝔽2) · Out(G) = C22 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C23 |
| prime radical | B(4,𝔽2) | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
129 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 129 conjugacy classes of subgroups — too many to draw. Only its 28 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:
(1,2)(3,9)(4,7)(5,8)(6,10)(11,15)(12,14)(13,16);(1,3,5,11)(2,6,8,13)(4,10,12,16)(7,9,14,15);(4,12)(10,16)Polycyclic presentation:
2^7|b^2=g,b^a=bd,c^a=ce,f^a=fg,e^b=efg,d^c=df,e^d=eg20 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2197, #33546, #33551, … |
| maximal normal subgroup of | C2×#2197, #33546, #33551, … |
| maximal quotient of | #12091, #12092, #12098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 929);