Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29751 |
| automorphism | Aut(G) order 8192 |
| 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×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
343 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,5)(3,6,7,8)(9,10,14,17,15,12,13,23)(11,16,20,24,21,22,19,18);(2,5)(6,8)(9,11)(10,16)(12,22)(13,19)(14,20)(15,21)(17,24)(18,23);(3,7)(6,8)(9,12)(10,15)(11,18)(13,23)(14,17)(16,19)(20,22)(21,24);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,17,12,23)(11,19,21,20)(16,24,22,18)Polycyclic presentation:
2^8|a^2=ef,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gh,d^c=de,f^c=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #273787, #273788, #273789, … |
| maximal normal subgroup of | #273787, #273788, #273789, … |
| maximal quotient of | #330473, #330474, #330477, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22851);