Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) = C2×#511819 (order 1024) · Inn(G) = #117 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #117 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #383 |
| prime radical | C23 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
81 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)(3,11,14,15)(4,7)(5,9)(6,10)(8,13,16,12);(2,7,10,9)(8,15,16,11);(1,3,4,12,6,14,5,13)(2,8,9,11,10,16,7,15)Polycyclic presentation:
2^7|a^2=f,b^2=de,c^2=d,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7225, #7226, #11836, … |
| maximal normal subgroup of | #7225, #7226, #11836, … |
| maximal quotient of | #8800, #8801, #8802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 573);