Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| 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 | #3013 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
430 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)(3,7)(4,6)(5,8)(9,10,13,19,15,21,14,20)(11,16,23,32,24,33,30,18)(12,25,27,36,29,35,28,17)(22,34,26,39,37,40,38,31);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,26)(13,23)(14,30)(15,24)(17,34)(18,20)(19,32)(21,33)(22,28)(25,39)(27,37)(29,38)(31,35)(36,40);(9,12,15,29)(10,17,21,36)(11,22,24,37)(13,28,14,27)(16,31,33,39)(18,34,32,40)(19,35,20,25)(23,38,30,26);(2,6)(3,8)(9,11)(10,18)(12,22)(13,30)(14,23)(15,24)(16,20)(17,34)(19,33)(21,32)(25,31)(26,28)(27,38)(29,37)(35,39)(36,40)Polycyclic presentation:
2^8|a^2=f,c^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6667751, #6667753, #6667757, … |
| maximal normal subgroup of | #6667751, #6667753, #6667757, … |
| maximal quotient of | #341983, #341984, #341985, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24037);