Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29751 |
| automorphism | Aut(G) order 16384 |
| 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 | #3038 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
356 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 356 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)(9,10,12,18,14,17,13,16)(11,15,20,23,21,22,19,24);(2,5)(6,8)(9,11)(10,15)(12,20)(13,19)(14,21)(16,24)(17,22)(18,23);(3,7)(6,8)(10,16)(11,19)(12,13)(15,22)(17,18)(20,21);(1,3,4,7)(2,6,5,8)(10,17)(15,22)(16,18)(23,24)Polycyclic presentation:
2^8|a^2=ef,d^2=e,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 | #273791, #273792, #273799, … |
| maximal normal subgroup of | #273791, #273792, #273799, … |
| maximal quotient of | #330471, #330475, #330479, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22849);