Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | Aut(G) = #72233209 (order 3072) · Inn(G) = #120 · Out(G) = GL(2,ℤ/4) |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C42 | C23 |
| prime radical | #127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
91 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7)(3,10)(4,5)(9,12)(11,13)(14,15)(18,22)(20,27)(23,31)(24,28)(25,29)(30,32);(1,2)(3,8)(4,12)(5,7)(6,9)(10,15)(11,16)(13,14)(17,18,21,24)(19,23,26,32)(20,28,27,22)(25,30,29,31);(1,3,6,11)(2,8,9,16)(4,10,5,13)(7,14,12,15)(17,19,21,26)(18,23,24,32)(20,29,27,25)(22,30,28,31)Polycyclic presentation:
2^7|b^2=g,c^2=f,d^2=f,e^2=fg,b^a=bd,c^a=ce,d^a=df,e^a=efg,d^b=df,e^b=eg,d^c=dg,e^c=efg20 conjugacy classes — view the full character table.
| Fitting subgroup of | #1658⋊C3, #11710972 |
| maximal subgroup of | #7524, C2×#1658, #18990, … |
| maximal normal subgroup of | #7524, C2×#1658, #18990, … |
| maximal quotient of | #7739, #7744, #7827, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 390);