Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) = #72262341 (order 8192) · Inn(G) = C2×C22⋊C4 · Out(G) = D4×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C24 | C23 |
| prime radical | C22×#41 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)Polycyclic presentation:
2^7|a^2=f,c^2=g,b^a=bd,c^a=ce,d^a=dg,f^b=fg32 conjugacy classes — view the full character table.
| Frattini subgroup of | #94522, #94523, #94622, … |
| maximal subgroup of | #7305, #7309, #7312, … |
| maximal normal subgroup of | #7305, #7309, #7312, … |
| maximal quotient of | #8300, #8302, #8303, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 514);