Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3212 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(4,11)(7,12)(8,13)(10,15)(14,16)(17,18)(19,26)(20,25)(21,23)(22,29)(24,27)(28,32)(30,31);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(18,23)(24,31)(25,29)(26,32);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,19)(18,24)(20,27)(21,28)(22,30)(23,31)(25,32)(26,29);(3,9)(7,14)(10,15)(12,16)(17,20,21,22)(18,25,23,29)(19,27,28,30)(24,32,31,26)Polycyclic presentation:
2^8|b^2=eh,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^c=dg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19224, #6353718, #6353719, … |
| maximal normal subgroup of | C2×#19224, #6353718, #6353719, … |
| maximal quotient of | #189577, #197774, #197775, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12212);