Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | Aut(G) = C2×#61451 (order 512) · Inn(G) = #459 · Out(G) = C23 |
| 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 | #509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
77 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)(3,11)(4,16)(5,20)(6,22)(7,12)(8,14)(9,13)(10,21)(15,29)(17,28)(18,32)(19,24)(23,25)(26,30)(27,31);(1,3,5,14,7,15,21,6)(2,8,10,11,12,22,20,29)(4,13,17,30,19,31,18,23)(9,24,25,28,27,16,26,32);(1,4)(2,9)(3,13)(5,17)(6,23)(7,19)(8,24)(10,25)(11,28)(12,27)(14,30)(15,31)(16,22)(18,21)(20,26)(29,32)Polycyclic presentation:
2^7|b^2=d,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 | C2×#2212, #33594, #33595, … |
| maximal normal subgroup of | C2×#2212, #33594, #33595, … |
| maximal quotient of | #12416, #12417, #12420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 944);