Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
463 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 463 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(5,6,8,9)(7,10,11,12)(13,14,17,21,18,22,32,38)(15,23,26,34,25,36,30,19)(16,27,28,35,29,37,31,20)(24,33,41,44,40,43,42,39);(1,2)(3,4)(6,9)(10,12)(13,15,18,25)(14,19,22,34)(16,24,29,40)(17,30,32,26)(20,33,35,43)(21,36,38,23)(27,39,37,44)(28,42,31,41);(2,4)(13,16,18,29)(14,20,22,35)(15,24,25,40)(17,31,32,28)(19,33,34,43)(21,37,38,27)(23,39,36,44)(26,42,30,41);(5,7,8,11)(6,10,9,12)(15,25)(19,34)(23,36)(24,40)(26,30)(33,43)(39,44)(41,42)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,d^2=efh,e^2=h,f^2=h,b^a=be,c^a=cf,c^b=cg,d^b=dh,e^b=eh,f^b=fh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #143157, #143161, #219495, … |
| maximal normal subgroup of | #143157, #143161, #219495, … |
| maximal quotient of | #244139, #244187, #244216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16802);