Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) = C2×#72213344 (order 4096) · Inn(G) = C2×D4 · Out(G) = C22×B(4,𝔽2) |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
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 | C23 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,7)(4,5)(6,8)(9,10,12,15,13,16,11,14);(2,5)(6,7)(10,14,16,15);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^7|a^2=f,b^2=def,c^2=e,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg44 conjugacy classes — view the full character table.
| Frattini subgroup of | #94548, #94549, #72272672 |
| maximal subgroup of | #7134, #7361, #8442, … |
| maximal normal subgroup of | #7134, #7361, #8442, … |
| maximal quotient of | #9870, #9871, #9876, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 726);