Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | Aut(G) = #72225648 (order 2048) · Inn(G) = #120 · Out(G) = D42 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C24 | C23 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
193 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 193 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6)(7,10)(8,15,16,11)(9,12)(13,17,19,20)(14,18);(1,2,3,4)(5,7,9,14)(6,10,12,18)(8,13,16,19)(11,17,15,20);(5,8)(6,11)(7,13)(9,16)(10,17)(12,15)(14,19)(18,20)Polycyclic presentation:
2^7|a^2=f,b^2=dfg,b^a=bd,c^a=ce,e^a=eg,e^b=eg,d^c=dg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #12754, #12760, #12761, … |
| maximal normal subgroup of | #12754, #12760, #12761, … |
| maximal quotient of | #8966, #8968, #9011, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 622);