Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | Aut(G) = #6628896 (order 512) · Inn(G) = C2×#99 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#99 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C2×D4 | C23 |
| prime radical | #534 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
140 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 42 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 — 3 generators on 16 points:
(1,2)(3,9)(4,12,13,7)(5,8)(6,11,14,16)(10,15);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16);(3,10)(6,14)(9,15)(11,16)Polycyclic presentation:
2^7|a^2=e,b^a=bd,d^a=df,f^a=fg,c^b=cg,e^b=ef,e^d=eg23 conjugacy classes — view the full character table.
| maximal subgroup of | #12705, #12706, #12725, … |
| maximal normal subgroup of | #12705, #12706, #12725, … |
| maximal quotient of | #11426, #11453, #11473, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 853);