Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3055 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
377 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)(3,6)(4,5)(7,8)(9,10,14,16,13,17,12,22)(11,18,21,24,20,15,19,23);(2,5)(6,8)(9,11,14,21,13,20,12,19)(10,15,16,23,17,18,22,24);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,17)(14,22)(18,20)(19,23)(21,24);(2,5)(6,8)(9,12,13,14)(10,16,17,22)(11,19,20,21)(15,23,18,24)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=eh,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25267, #6441696, #6441698, … |
| maximal normal subgroup of | C2×#25267, #6441696, #6441698, … |
| maximal quotient of | #285632, #285633, #285636, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18255);