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, prime radical | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
277 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 277 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) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(3,9)(4,12)(7,10)(8,13)(15,16)(17,18,21,24)(19,25,27,32)(20,28,30,23)(22,29,31,26);(1,2,5,8)(3,10,9,15)(4,13,12,6)(7,14,16,11);(17,19,21,27)(18,22,24,31)(20,29,30,26)(23,32,28,25);(1,3,5,9)(2,7,8,16)(4,14,12,11)(6,15,13,10)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=h,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 | #492099, #492102, #492107, … |
| maximal normal subgroup of | #492099, #492102, #492107, … |
| maximal quotient of | Q16×#103, #220819, #220825, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12996);