Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,14,18)(11,20,23,17)(12,25,27,33)(13,29,31,28)(15,32,16,19)(21,30,40,22)(24,37,34,38)(26,39,36,35);(2,5)(3,8)(9,11,14,23)(10,17,18,20)(12,26,27,36)(13,30,31,22)(15,24,16,34)(19,37,32,38)(21,28,40,29)(25,35,33,39);(9,12,14,27)(10,13,18,31)(11,21,23,40)(15,29,16,28)(17,35,20,39)(19,33,32,25)(22,37,30,38)(24,36,34,26);(1,3,4,8)(2,6,5,7)(9,13,14,31)(10,12,18,27)(11,22,23,30)(15,33,16,25)(17,36,20,26)(19,29,32,28)(21,37,40,38)(24,35,34,39)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #6670994, #6670999, #6671006, … |
| maximal normal subgroup of | #6670994, #6670999, #6671006, … |
| maximal quotient of | #340447, #340450, #340451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23932);