Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) = #72262696 (order 4096) · Inn(G) = C2×C22⋊C4 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
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 | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,4)(5,6,7,10,8,9,11,12);(1,2,3,4);(6,9)(7,11)Polycyclic presentation:
2^7|a^2=f,b^2=d,f^2=g,b^a=bd,c^a=ce,e^a=eg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7236, #7321, #8353, … |
| maximal normal subgroup of | #7236, #7321, #8353, … |
| maximal quotient of | #8557, #8559, #8561, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 532);