Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) = #72251325 (order 4096) · Inn(G) = C2×C22⋊C4 · Out(G) = C2×#392 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,9,12,8,11,15,17)(7,10,14,19,13,18,16,20);(5,7,9,16,8,13,15,14)(6,10,17,19,11,18,12,20);(1,2,3,4)(6,11)(10,18)(12,17)(19,20)Polycyclic presentation:
2^7|a^2=f,b^2=df,c^2=eg,f^2=g,b^a=bd,c^a=ce,d^a=dg,d^b=dg,f^b=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #8483, #8484, #10496, … |
| maximal normal subgroup of | #8483, #8484, #10496, … |
| maximal quotient of | #9651, #9652, #9687, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 699);