Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
491 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 491 conjugacy classes of subgroups — too many to draw. Only its 151 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)(9,10,13,16)(11,17,20,24)(12,21,18,15)(14,22,23,19);(5,6,7,8)(9,11,18,22,13,20,12,19)(10,14,21,24,16,23,15,17);(2,4)(6,8);(9,11,18,22,13,20,12,19)(10,14,21,24,16,23,15,17)Polycyclic presentation:
2^8|a^2=h,b^2=egh,d^2=eh,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg58 conjugacy classes — view the full character table.
| maximal subgroup of | #143698, #143700, #143746, … |
| maximal normal subgroup of | #143698, #143700, #143746, … |
| maximal quotient of | #318008, #318013, #318014, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21216);