Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) = C2×#10576928 (order 1024) · Inn(G) = D8 · Out(G) = C22×#41 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D8 |
| commutator | C8 | C42 |
| Frattini, cosocle | C22×C8 | C22 |
| socle | C22 | #102 |
| prime radical | C23 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,3,4)(5,6)(7,16)(8,21)(9,24)(10,26)(11,15)(12,17)(13,23)(14,22)(18,28)(19,36)(20,32)(25,29)(27,35)(30,34)(31,33);(5,7,9,18,22,33,23,34,11,20,8,17,10,19,25,35)(6,12,13,16,26,30,24,36,15,28,29,32,14,27,21,31)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=fg,e^2=fg,f^2=g,b^a=bc,c^a=cf,e^a=efg,f^a=fg44 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1382, #12561, C4×#359, … |
| maximal normal subgroup of | C2×#1382, #12561, C4×#359, … |
| maximal quotient of | #7155, #7171, #7172, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 114);