Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
524 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 524 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) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,3)(9,10,13,17,14,18,27,34)(11,15,21,30,20,29,37,40)(12,22,24,32,25,33,26,16)(19,35,36,31,23,39,38,28);(1,2,4,3)(5,6,7,8)(9,11)(10,15)(12,23)(13,21)(14,20)(16,31)(17,30)(18,29)(19,25)(22,39)(24,38)(26,36)(27,37)(28,32)(33,35)(34,40);(6,8)(9,12,14,25)(10,16,18,32)(11,19,20,23)(13,26,27,24)(15,28,29,31)(17,33,34,22)(21,38,37,36)(30,39,40,35);(1,3,4,2)(11,20)(15,29)(19,23)(21,37)(28,31)(30,40)(35,39)(36,38)Polycyclic presentation:
2^8|a^2=f,b^2=egh,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28456, #6465543, #6465553, … |
| maximal normal subgroup of | C2×#28456, #6465543, #6465553, … |
| maximal quotient of | #321773, #321774, #321775, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21444);