Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
353 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,5)(3,7,8,6)(9,10,12,14)(11,13,15,16)(17,18,23,20)(19,24,21,22);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,18,23,21,24,20);(11,15)(13,16)(17,20,22,19,23,18,24,21);(1,3,4,8)(2,6,5,7)(17,21,22,20,23,19,24,18)Polycyclic presentation:
2^8|a^2=efgh,b^2=f,c^2=f,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28017, #6457868, #6457881, … |
| maximal normal subgroup of | C2×#28017, #6457868, #6457881, … |
| maximal quotient of | #360930, #362678, #367309, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21005);