Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,12,16,13,17,22,21)(11,14,18,15,20,24,19,23);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,18)(13,20)(15,16)(17,24)(19,22)(21,23);(1,4,5,8)(2,6,3,7)(9,10)(11,15)(12,21)(13,17)(14,18)(16,22)(19,24)(20,23);(2,3)(6,7)(9,11)(10,15)(12,19)(13,20)(14,16)(17,23)(18,22)(21,24)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #6669710, #6669712, #6669714, … |
| maximal normal subgroup of | #6669710, #6669712, #6669714, … |
| maximal quotient of | #344208, #344211, #344212, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24172);