Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | Aut(G) = C2×#72224760 (order 3072) · Inn(G) = #127 · Out(G) = C22×S4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #127 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #392 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
175 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)(5,22)(6,10)(7,11)(8,14)(9,20)(13,30)(15,25)(16,24)(18,29)(19,28)(21,27)(23,31)(26,32);(1,3)(2,8)(4,18)(5,15)(6,14)(7,16)(9,26)(10,24)(11,25)(12,22)(13,19)(17,23)(20,31)(21,32)(27,30)(28,29);(1,4)(2,9)(3,13)(5,19)(6,20)(7,21)(8,23)(10,27)(11,28)(12,29)(14,18)(15,32)(16,31)(17,22)(24,26)(25,30)Polycyclic presentation:
2^7|d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cf,d^b=dg,f^b=fg,e^c=eg,f^c=fg26 conjugacy classes — view the full character table.
| Fitting subgroup of | #2017⋊C3, #11710961, #11710962, … |
| maximal subgroup of | #7406, #7407, #12117, … |
| maximal normal subgroup of | #7406, #7407, #12117, … |
| maximal quotient of | #9974, #9975, #9976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 749);