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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
310 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 103 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:
(1,2,7,4)(3,11,14,16)(5,10,6,9)(8,15,12,13)(17,18)(19,22)(20,23)(21,25)(24,28)(26,29)(27,31)(30,32);(1,3)(2,8)(4,12)(5,13)(6,15)(7,14)(9,11)(10,16)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30);(1,4,7,2)(3,12,14,8)(5,10,6,9)(11,13,16,15)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(1,5)(2,9)(4,10)(6,7)(11,16)(13,15)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^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 | #491905, #491908, #491913, … |
| maximal normal subgroup of | #491905, #491908, #491913, … |
| maximal quotient of | #220808, #220828, #220830, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12977);