Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) = #72218077 (order 1024) · Inn(G) = #99 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #99 |
| commutator | C2×C4 | C42 |
| Frattini, cosocle | C22×Q8 | C22 |
| prime radical | C23 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 20 points:
(5,6,8,12,9,13,17,20)(7,11,14,19,15,10,16,18);(1,2,3,4)(6,10,13,11)(12,19,20,18)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=g,d^2=g,b^a=bc,c^a=cf,e^a=eg,f^a=fg,d^b=dfg,d^c=dg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7404, #7405, #7409, … |
| maximal normal subgroup of | #7404, #7405, #7409, … |
| maximal quotient of | #7189, #7206, #7215, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 126);