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×D8 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
303 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 303 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,9)(3,10,13,14)(4,5,15,8)(7,16,11,12)(17,18)(19,22)(20,23)(21,25)(24,28)(26,29)(27,31)(30,32);(1,3,6,13)(2,7,9,11)(4,14,15,10)(5,16,8,12)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30);(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(1,4,6,15)(2,8,9,5)(3,11,13,7)(10,16,14,12)Polycyclic presentation:
2^8|a^2=g,b^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 | #492097, #492100, #492105, … |
| maximal normal subgroup of | #492097, #492100, #492105, … |
| maximal quotient of | #220811, D8×#103, #220816, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12981);