Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C25 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
527 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 527 conjugacy classes of subgroups — too many to draw. Only its 117 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10)(3,11,13,27)(4,15,17,29)(5,8,19,23)(7,20,22,12)(9,24,26,18)(14,25,28,32)(16,21,30,31);(1,3,5,14)(2,7,8,21)(4,16,18,12)(6,13,19,28)(9,25,15,11)(10,22,23,31)(17,30,24,20)(26,32,29,27);(4,17)(9,26)(11,27)(12,20)(15,29)(16,30)(18,24)(25,32);(2,8)(7,21)(9,15)(10,23)(11,25)(22,31)(26,29)(27,32)Polycyclic presentation:
2^8|a^2=h,b^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fh,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #246331, #246332, #246339, … |
| maximal normal subgroup of | #246331, #246332, #246339, … |
| maximal quotient of | #165879, #165889, #165916, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11126);