Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19621 |
| 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 | #3184 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
360 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)(3,9)(4,13)(5,7)(6,12)(8,10)(11,15)(14,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,31)(29,32);(1,3,10,16,5,11,4,12)(2,6,13,15,7,14,8,9);(2,7)(6,14)(8,13)(9,15)(17,19,20,27,22,29,21,28)(18,23,30,32,25,31,24,26);(3,11)(6,14)(9,15)(12,16)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,26,31,32)Polycyclic presentation:
2^8|b^2=eg,c^2=fgh,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dg52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#19650, #6357136, #6357139, … |
| maximal normal subgroup of | C2×#19650, #6357136, #6357139, … |
| maximal quotient of | #191443, #206131, #206134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12638);