Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) = C2×#72217072 (order 8192) · Inn(G) = C2×D4 · Out(G) = C2×#23900 |
| 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×C4⋊C4 |
| prime radical | C23 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,9,12,10,13,8,18)(7,14,16,19,17,20,15,11);(5,7,8,15,10,17,9,16)(6,11,12,14,13,19,18,20);(1,2,3,4)Polycyclic presentation:
2^7|a^2=f,b^2=d,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 | #94569, #94570, #94573, … |
| maximal subgroup of | #7270, #7272, #7273, … |
| maximal normal subgroup of | #7270, #7272, #7273, … |
| maximal quotient of | #8805, #8806, #8807, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 580);