Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D82 |
| automorphism | Aut(G) order 16384 |
| 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 | |
283 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 283 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:
(2,7)(3,5)(4,11)(8,12)(9,13)(15,16)(17,18,21,24)(19,25,27,32)(20,28,30,23)(22,29,31,26);(1,2,6,9)(3,7,5,13)(4,12,11,15)(8,14,16,10);(1,3,6,5)(2,7,9,13)(4,10,11,14)(8,15,16,12)(17,19,21,27)(18,22,24,31)(20,29,30,26)(23,32,28,25);(1,4,6,11)(2,8,9,16)(3,10,5,14)(7,15,13,12)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=gh,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 | #123282, #123292, #123304, … |
| maximal normal subgroup of | #123282, #123292, #123304, … |
| maximal quotient of | #220819, #220838, Q16×#107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12998);