Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | Aut(G) = #72218133 (order 2048) · Inn(G) = #120 · Out(G) = D42 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C24 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
319 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 319 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12);(9,12);(2,4)(5,7)(6,9)(8,11)(10,12)Polycyclic presentation:
2^7|b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^c=dg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7394, #12090, #12091, … |
| maximal normal subgroup of | #7394, #12090, #12091, … |
| maximal quotient of | #9914, #9915, #9916, … |
| automorphism group of | C4×C8, #97 |
| outer automorphism group of | C4×C8, C2×C22⋊C4, #322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 753);