Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| 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 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
442 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)(3,7)(4,6)(5,8)(9,10,14,16,13,17,12,23)(11,18,22,24,21,15,20,19);(1,3,4,8)(2,5,6,7)(9,11,13,21)(10,15,17,18)(12,20,14,22)(16,19,23,24);(9,10)(11,19)(12,16)(13,17)(14,23)(15,22)(18,20)(21,24);(2,6)(5,7)(9,12,13,14)(10,16,17,23)(11,20,21,22)(15,19,18,24)Polycyclic presentation:
2^8|a^2=f,b^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24524, #6452025, #6452028, … |
| maximal normal subgroup of | C2×#24524, #6452025, #6452028, … |
| maximal quotient of | #274079, #274086, #274091, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17512);