Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C23 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
291 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 291 conjugacy classes of subgroups — too many to draw. Only its 115 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,16)(14,15)(17,18,22,25)(19,26,24,21)(20,27,28,32)(23,29,31,30);(2,6)(7,15)(8,13)(9,16)(17,19,21,18,22,24,26,25)(20,27,29,23,28,32,30,31);(1,3,4,10)(2,7,6,15)(5,14,12,11)(8,16,13,9)(17,20,22,28)(18,23,25,31)(19,27,24,32)(21,29,26,30);(20,28)(23,31)(27,32)(29,30)Polycyclic presentation:
2^8|a^2=g,b^2=egh,c^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^c=dg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #263067, #263072, #263107, … |
| maximal normal subgroup of | #263067, #263072, #263107, … |
| maximal quotient of | #189581, #197767, #197770, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12203);