Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19713 |
| 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 | #3207 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(11,17)(12,19)(14,20)(16,22)(18,23)(21,24);(1,2,5,7,6,8,4,3)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24);(1,3,4,8,6,7,5,2)(9,11,12,17,13,18,19,23)(10,14,16,20,15,21,22,24);(1,4,6,5)(2,7,8,3)(10,15)(14,21)(16,22)(20,24)Polycyclic presentation:
2^8|b^2=eg,c^2=fg,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dh,d^c=dg40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19725, #6357363, #6357371, … |
| maximal normal subgroup of | C2×#19725, #6357363, #6357371, … |
| maximal quotient of | #191524, #206259, #206264, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12713);