Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2540 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
559 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)(9,10,14,18)(11,19,15,16)(12,22,13,23)(17,21,24,20);(1,2)(3,4)(5,6)(7,8)(9,11,14,15)(10,16,18,19)(12,20,13,21)(17,22,24,23);(6,8)(9,12,14,13)(10,17,18,24)(11,20,15,21)(16,22,19,23);(2,4)(9,13,14,12)(10,17,18,24)(11,21,15,20)(16,22,19,23)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=deh,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6632023, #6632033, #6632041, … |
| maximal normal subgroup of | #6632023, #6632033, #6632041, … |
| maximal quotient of | #332975, #332976, #332977, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23506);