Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | Aut(G) = C2×D4×AGL(1,ℤ/8) (order 512) · Inn(G) = D42 · Out(G) = C23 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
227 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 227 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,9)(5,22)(6,11)(7,15)(8,20)(10,23)(13,25)(14,31)(16,27)(17,26)(18,24)(19,29)(21,28)(30,32);(1,3)(2,7)(4,18)(5,16)(6,17)(8,25)(9,30)(10,26)(11,27)(12,22)(13,21)(14,19)(15,23)(20,32)(24,29)(28,31);(1,4)(2,8)(3,13)(5,19)(6,21)(7,24)(9,22)(10,28)(11,29)(12,31)(14,15)(16,32)(17,18)(20,23)(25,27)(26,30);(1,4)(2,9)(3,14)(5,20)(6,21)(7,25)(8,22)(10,29)(11,28)(12,31)(13,15)(16,18)(17,32)(19,23)(24,27)(26,30)Polycyclic presentation:
2^7|e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,c^b=cg,d^b=de,e^b=eg,f^c=fg,e^d=eg29 conjugacy classes — view the full character table.
| maximal subgroup of | #33854, #33855, #33857, … |
| maximal normal subgroup of | #33854, #33855, #33857, … |
| maximal quotient of | #19883, #19886, #19889, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2022);