Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
288 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 97 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:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,22)(12,21)(13,19)(14,29)(15,18)(16,24)(17,25)(20,31)(23,33)(26,36)(27,37)(28,32)(30,38)(34,40)(35,39);(1,3)(2,5)(4,8)(6,7)(9,11,13,24)(10,16,19,22)(12,26,15,23)(14,25,28,39)(17,29,35,32)(18,36,21,33)(20,34,37,38)(27,40,31,30);(3,8)(5,7)(9,11)(10,17)(12,23)(13,24)(14,30)(15,26)(16,32)(18,34)(19,35)(20,36)(21,38)(22,29)(25,31)(27,39)(28,40)(33,37);(2,6)(5,7)(9,12,13,15)(10,18,19,21)(11,23,24,26)(14,27,28,31)(16,33,22,36)(17,34,35,38)(20,32,37,29)(25,30,39,40)Polycyclic presentation:
2^8|b^2=h,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,c^b=ce,d^b=dh,f^c=fg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #151178, #151181, #151487, … |
| maximal normal subgroup of | #151178, #151181, #151487, … |
| maximal quotient of | #496272, #496678, #496826, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26757);