Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) = C2×#63011 (order 512) · Inn(G) = #120 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #120 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #385 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,6,10,7,11,5,13)(3,12,9,8,15,16,14,4);(1,3,6,9,7,15,5,14)(2,8,10,16,11,4,13,12);(1,4)(2,9)(3,13)(5,16)(6,12)(7,8)(10,15)(11,14)Polycyclic presentation:
2^7|a^2=f,b^2=f,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,e^c=eg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1861, #20867, #21477, … |
| maximal normal subgroup of | C2×#1861, #20867, #21477, … |
| maximal quotient of | #8839, #8840, #8841, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 593);