Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19733 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
335 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,9)(4,11)(6,12)(8,14)(10,15)(13,16)(17,18,22,25)(19,26,24,21)(20,28,29,27)(23,31,32,30);(1,2)(3,6)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(17,19,21,18,22,24,26,25)(20,27,30,23,29,28,31,32);(1,3,4,9,5,10,11,15)(2,6,8,12,7,13,14,16)(17,20,26,31,22,29,21,30)(18,23,19,27,25,32,24,28);(2,7)(6,13)(8,14)(12,16)(20,29)(23,32)(27,28)(30,31)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=fg,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dgh,d^c=dg40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19738, #6357385, #6357389, … |
| maximal normal subgroup of | C2×#19738, #6357385, #6357389, … |
| maximal quotient of | #191529, #206275, #206278, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12726);