Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) = #72234528 (order 4096) · Inn(G) = C22⋊C4 · Out(G) = #12687 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22⋊C4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C42 | C23 |
| socle | C22 | C2×C22⋊C4 |
| prime radical | C23 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18)(19,26)(20,24)(21,29)(22,25)(23,27)(28,31)(30,32);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,22,28)(18,23,25,32)(20,27,21,30)(24,31,29,26);(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)Polycyclic presentation:
2^7|a^2=e,b^2=f,c^2=g,d^2=g,e^2=fg,b^a=bd,d^a=df,d^b=dg,e^b=efg38 conjugacy classes — view the full character table.
| maximal subgroup of | #8242, #8244, #8246, … |
| maximal normal subgroup of | #8242, #8244, #8246, … |
| maximal quotient of | #7564, #7568, #7646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 276);