Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | Aut(G) = C26×#442 (order 4096) · Inn(G) = C22×D8 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
303 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 99 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,17)(13,16)(14,27)(15,21)(18,26)(19,25)(22,28)(23,30)(24,32)(29,38)(31,36)(33,40)(34,39)(35,37);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,21)(13,22)(14,23)(16,31)(17,20)(18,29)(19,30)(24,36)(25,37)(26,39)(27,35)(28,32)(33,34)(38,40);(3,8)(5,7)(10,16)(11,21)(13,26)(14,25)(15,28)(17,32)(18,19)(20,31)(22,38)(24,40)(27,33)(29,35)(30,34)(36,39);(2,6)(5,7)(10,17)(11,21)(16,32)(18,33)(19,27)(22,36)(23,37)(38,39)Polycyclic presentation:
2^8|b^2=eh,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 | C2×#33790, #7623352, #7623361, … |
| maximal normal subgroup of | C2×#33790, #7623352, #7623361, … |
| maximal quotient of | #496371, #496488, #496744, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26778);