Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) = #72236881 (order 1024) · Inn(G) = #121 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #121 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #380 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12)(4,17,19,24)(5,20)(6,10)(7,11)(8,14)(9,18,25,21)(13,22,29,32)(15,23)(16,27)(26,28,31,30);(1,3,5,14,7,16,6,15)(2,8,10,12,11,23,20,27)(4,13,18,28,19,29,21,30)(9,22,24,26,25,32,17,31);(1,4)(2,9)(3,13)(5,18)(6,21)(7,19)(8,22)(10,24)(11,25)(12,26)(14,28)(15,30)(16,29)(17,20)(23,32)(27,31)Polycyclic presentation:
2^7|a^2=f,b^2=d,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^c=eg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #12747, #12750, #12751, … |
| maximal normal subgroup of | #12747, #12750, #12751, … |
| maximal quotient of | #8604, #8605, #8606, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 546);