Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | Aut(G) = #72236721 (order 2048) · Inn(G) = C2×#120 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#120 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
368 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,7)(4,5)(8,15)(9,12)(10,13)(14,16);(1,2)(3,8)(4,7)(5,12)(6,9)(10,14)(11,16)(13,15);(1,3)(2,8)(4,10)(5,13)(6,11)(7,14)(9,16)(12,15);(1,4,6,5)(2,7,9,12)(3,10,11,13)(8,14,16,15)Polycyclic presentation:
2^7|d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fg,e^c=eg29 conjugacy classes — view the full character table.
| maximal subgroup of | #13353, #13354, #13360, … |
| maximal normal subgroup of | #13353, #13354, #13360, … |
| maximal quotient of | #18135, #18141, #18145, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1753);