Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#135 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
993 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)(3,4)(9,10)(11,18)(12,20)(13,16)(14,23)(15,17)(19,22)(21,24);(2,4)(5,6)(7,8)(9,11,15,21)(10,12,17,22)(13,20,14,19)(16,24,23,18);(6,8)(9,12)(10,11)(13,18)(14,24)(15,22)(16,19)(17,21)(20,23);(9,13,15,14)(10,16,17,23)(11,19,21,20)(12,18,22,24)Polycyclic presentation:
2^8|b^2=h,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=cg,d^b=dh,e^b=eh,f^b=fh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #376197, #381479, #381888, … |
| maximal normal subgroup of | #376197, #381479, #381888, … |
| maximal quotient of | #232220, #232226, #232228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16415);