Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) = #72231110 (order 2048) · Inn(G) = C22⋊C4 · Out(G) = #3003 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22⋊C4 |
| commutator | C2×C4 | C42 |
| Frattini, cosocle | C23×C4 | C22 |
| socle | C22 | #95 |
| prime radical | C24 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,4,7,5,8,3,6)(9,10)(11,15)(12,14)(13,16);(2,6,8,7)(9,11)(10,13,14,15)(12,16)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=f,d^2=f,b^a=bc,c^a=cf,e^a=efg,c^b=cg38 conjugacy classes — view the full character table.
| maximal subgroup of | #7396, #7432, C2×#1297, … |
| maximal normal subgroup of | #7396, #7432, C2×#1297, … |
| maximal quotient of | #7026, #7028, #7035, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 29);