Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×#62773 (order 512) · Inn(G) = D4 · Out(G) = C22×#41 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | D4 |
| commutator | C4 | C4×C8 |
| Frattini, cosocle | C22×C8 | C22 |
| socle | C22 | C22⋊C8 |
| prime radical | C23 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 36 points:
(5,6,9,14,10,15,24,32,11,16,25,33,8,13,22,30)(7,17,19,34,18,28,35,31,20,29,23,36,21,12,26,27);(1,2,3,4)(5,7)(6,12)(8,21)(9,23)(10,18)(11,20)(13,29)(14,31)(15,17)(16,28)(19,25)(22,35)(24,26)(27,33)(30,34)(32,36)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=g,d^2=f,f^2=g,b^a=bc,c^b=cg,d^b=dg56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1379, #12554, #12597, … |
| maximal normal subgroup of | C2×#1379, #12554, #12597, … |
| maximal quotient of | #7141, #7144, #7147, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 111);