Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| 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 | C23 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
465 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 465 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) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,15)(12,22)(13,25)(14,18)(16,24)(17,28)(19,36)(20,34)(21,31)(23,35)(26,30)(27,33)(29,38)(32,39)(37,40);(1,3,5,4)(2,6,8,7)(9,11,14,21)(10,15,18,31)(12,19,23,37)(13,26,28,20)(16,32,33,29)(17,34,25,30)(22,36,35,40)(24,39,27,38);(9,12,14,23)(10,16,18,33)(11,19,21,37)(13,27,28,24)(15,29,31,32)(17,35,25,22)(20,39,26,38)(30,40,34,36);(1,4,5,3)(2,7,8,6)(12,23)(16,33)(19,37)(22,35)(24,27)(29,32)(36,40)(38,39)Polycyclic presentation:
2^8|a^2=e,b^2=eh,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,d^c=dh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #245727, #245734, #245748, … |
| maximal normal subgroup of | #245727, #245734, #245748, … |
| maximal quotient of | #265464, #265467, #265472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17307);