Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19072 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(4,5)(9,10)(12,13)(14,15)(18,23)(20,21)(24,31)(25,29)(26,27)(30,32);(1,2)(3,8)(4,9)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,22,25)(19,24,28,32)(20,23,21,29)(26,31,27,30);(1,3,6,13)(2,8,10,16)(4,11,5,12)(7,14,9,15)(17,19,22,28)(18,24,25,32)(20,27,21,26)(23,30,29,31);(1,4,6,5)(2,9,10,7)(3,12,13,11)(8,14,16,15)(17,20,22,21)(18,23,25,29)(19,26,28,27)(24,31,32,30)Polycyclic presentation:
2^8|b^2=h,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fgh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fgh40 conjugacy classes — view the full character table.
| Fitting subgroup of | #19083⋊C3 |
| maximal subgroup of | #153497, #249744, #249757, … |
| maximal normal subgroup of | #153497, #249744, #249757, … |
| maximal quotient of | #168132, #168172, #168189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12071);