Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
545 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 545 conjugacy classes of subgroups — too many to draw. Only its 179 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(9,10,12,15,13,16,21,24)(11,17,18,22,19,23,20,14);(1,2,4,3)(5,6,7,8);(6,8)(9,11,13,19)(10,14,16,22)(12,20,21,18)(15,23,24,17);(1,3,4,2)Polycyclic presentation:
2^8|a^2=f,b^2=eg,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #101730, #101750, #101847, … |
| maximal normal subgroup of | #101730, #101750, #101847, … |
| maximal quotient of | #321249, #321257, #321264, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21428);