Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
516 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 128 |
A minimal faithful permutation representation — 4 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,22)(15,19,24,33)(16,26,28,36)(17,29,31,21)(20,30,35,27)(23,37,38,44)(25,41,40,34)(32,42,43,39);(1,2)(3,4)(6,9)(10,12)(13,15,18,24)(14,19,22,33)(16,23,28,38)(17,25,31,40)(20,32,35,43)(21,34,29,41)(26,37,36,44)(27,39,30,42);(2,4)(7,11)(10,12)(13,16,18,28)(14,20,22,35)(15,23,24,38)(17,30,31,27)(19,32,33,43)(21,36,29,26)(25,42,40,39)(34,44,41,37);(5,7,8,11)(6,10,9,12)(15,24)(19,33)(23,38)(25,40)(32,43)(34,41)(37,44)(39,42)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,d^b=dh,d^c=de,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #269167, #269169, #269207, … |
| maximal normal subgroup of | #269167, #269169, #269207, … |
| maximal quotient of | #328422, #328425, #328428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22662);