Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) = #72234840 (order 1024) · Inn(G) = #99 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #99 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C2×Q8 | C23 |
| socle | C2 | C2×#99 |
| prime radical | #532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
125 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,5,10,6,11,3,8)(4,9,12,16,13,7,14,15);(2,7)(4,13)(8,15)(9,11)(10,16)(12,14);(1,3,6,5)(2,8,11,10)(4,12,13,14)(7,15,9,16)Polycyclic presentation:
2^7|a^2=e,c^2=g,d^2=g,e^2=g,b^a=bd,d^a=df,f^a=fg,d^b=dg,e^b=efg,e^d=eg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7484, #7488, #7492, … |
| maximal normal subgroup of | #7484, #7488, #7492, … |
| maximal quotient of | #11431, #11447, #11475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 864);