Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D82 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×D4 | C2×D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
311 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 311 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7)(3,5,8,6)(10,14)(11,17)(12,20)(15,18)(16,21)(23,24);(9,10,13,16)(11,18,17,23)(12,21,20,14)(15,22,24,19);(2,5)(3,8)(6,7);(9,11,13,17)(10,15,16,24)(12,22,20,19)(14,23,21,18)Polycyclic presentation:
2^8|a^2=h,b^2=g,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,d^b=de,e^b=eg,f^c=fh,e^d=eg49 conjugacy classes — view the full character table.
| maximal subgroup of | #491782, #491790, #491799, … |
| maximal normal subgroup of | #491782, #491790, #491799, … |
| maximal quotient of | Q16×#102, #220819, #220821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12971);