Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3489 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
625 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)(11,12)(13,17)(14,18)(15,19)(16,23)(20,22)(21,24);(2,5)(6,8)(9,11)(10,12)(13,20)(14,16)(15,21)(17,22)(18,23)(19,24);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,17)(13,22)(14,23)(15,24)(18,21)(19,20);(2,5)(6,8)(9,13,15,14)(10,17,19,18)(11,20,21,16)(12,22,24,23)Polycyclic presentation:
2^8|c^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,f^c=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24028, #6440370, #6440372, … |
| maximal normal subgroup of | C2×#24028, #6440370, #6440372, … |
| maximal quotient of | #254827, #254830, #254837, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17016);