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 | C23 | #127 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12)(3,9,13,16)(4,8,14,15)(5,11,6,10)(19,25)(20,27)(22,28)(24,30)(26,31)(29,32);(1,3,6,14,7,13,5,4)(2,8,11,16,12,15,10,9)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(1,4,5,13,7,14,6,3)(2,9,10,15,12,16,11,8)(17,19,20,25,21,26,27,31)(18,22,24,28,23,29,30,32);(3,13)(4,14)(8,15)(9,16)(18,23)(22,29)(24,30)(28,32)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=fg,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×#19720, #6357363, #6357371, … |
| maximal normal subgroup of | C2×#19720, #6357363, #6357371, … |
| maximal quotient of | #191524, #206261, #206263, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12708);