Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,22)(12,28)(13,32)(14,20)(15,21)(16,25)(17,24)(18,33)(19,29)(23,39)(26,36)(27,38)(30,35)(31,40)(34,37);(2,5)(6,8)(9,11,14,25,15,27,29,26)(10,16,19,22,21,36,20,38)(12,13,24,33,31,34,30,23)(17,37,28,18,35,32,40,39);(1,3,4,7)(2,6,5,8)(9,12,29,30,15,31,14,24)(10,17,20,40,21,35,19,28)(11,23,26,34,27,33,25,13)(16,18,38,37,36,39,22,32);(2,5)(6,8)(9,13,29,23,15,34,14,33)(10,18,20,37,21,39,19,32)(11,24,26,12,27,30,25,31)(16,35,38,28,36,17,22,40)Polycyclic presentation:
2^8|b^2=f,c^2=eh,d^2=fh,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27293, #6440839, #6440841, … |
| maximal normal subgroup of | C2×#27293, #6440839, #6440841, … |
| maximal quotient of | #313710, #313711, #313714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20281);