Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) = C2×#72210600 (order 2048) · Inn(G) = C2×D4 · Out(G) = #2021 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C22 | C2×C22⋊C4 |
| prime radical | #525 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
221 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 221 conjugacy classes of subgroups — too many to draw. Only its 60 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)(5,6,8,11,9,12,7,10);(6,10,12,11);(2,4)Polycyclic presentation:
2^7|a^2=f,b^2=df,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg44 conjugacy classes — view the full character table.
| maximal subgroup of | #8346, #8347, C2×#1896, … |
| maximal normal subgroup of | #8346, #8347, C2×#1896, … |
| maximal quotient of | #9071, #9072, #9077, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 628);