Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
725 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 725 conjugacy classes of subgroups — too many to draw. Only its 143 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(11,18)(15,23)(19,24)(20,21);(2,4)(9,10,13,17)(11,19,18,24)(12,16,14,22)(15,20,23,21);(6,8)(9,11,13,18)(10,15,17,23)(12,21,14,20)(16,24,22,19);(2,4)(9,12,13,14)(10,16,17,22)(11,20,18,21)(15,19,23,24)Polycyclic presentation:
2^8|a^2=fh,b^2=h,c^2=h,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #273371, #273388, #273407, … |
| maximal normal subgroup of | #273371, #273388, #273407, … |
| maximal quotient of | #267205, #267206, #267207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17365);