Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | Aut(G) = #72237138 (order 1024) · Inn(G) = #459 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| prime radical | #135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
75 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,8,14)(3,12,17,23)(4,18,20,31)(5,13,7,11)(6,9,16,24)(10,25,27,15)(19,32,22,28)(21,29,30,26);(1,3,7,16,8,17,5,6)(2,9,13,23,14,24,11,12)(4,15,22,30,20,25,19,21)(10,18,29,32,27,31,26,28);(1,4)(2,10)(3,15)(5,19)(6,21)(7,22)(8,20)(9,18)(11,26)(12,28)(13,29)(14,27)(16,30)(17,25)(23,32)(24,31)Polycyclic presentation:
2^7|a^2=g,b^2=dg,d^2=g,e^2=fg,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ef,f^a=fg,e^c=ef,f^c=fg23 conjugacy classes — view the full character table.
| maximal subgroup of | #12312, #12313, #12315, … |
| maximal normal subgroup of | #12312, #12313, #12315, … |
| maximal quotient of | #12422, #12423, #12426, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 946);