Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
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 | |
419 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 143 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:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,24)(15,25,30,43)(16,31,32,22)(17,21,34,40)(18,35,26,23)(20,36,39,29)(27,42,44,37)(28,41,33,38);(6,9)(10,12)(13,15,26,36,19,30,18,29)(14,20,35,43,24,39,23,25)(16,27,34,28,32,44,17,33)(21,37,22,41,40,42,31,38);(1,2,3,4)(13,16,26,34,19,32,18,17)(14,21,35,22,24,40,23,31)(15,27,36,28,30,44,29,33)(20,37,43,41,39,42,25,38);(5,7,8,11)(6,10,9,12)(13,17,18,32,19,34,26,16)(14,22,23,21,24,31,35,40)(15,28,29,27,30,33,36,44)(20,38,25,42,39,41,43,37)Polycyclic presentation:
2^8|a^2=h,b^2=gh,c^2=fh,d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=gh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #125301, #125424, #140996, … |
| maximal normal subgroup of | #125301, #125424, #140996, … |
| maximal quotient of | #352642, #352647, #352651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26237);