Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29748 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
339 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 339 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(1,2,4,5)(3,6,7,8)(9,10,15,20,13,18,14,19)(11,21,30,32,23,16,24,36)(12,25,29,17,27,35,28,34)(22,38,40,37,39,33,26,31);(2,5)(6,8)(9,11)(10,16)(12,26)(13,23)(14,24)(15,30)(17,33)(18,21)(19,32)(20,36)(22,29)(25,37)(27,40)(28,39)(31,35)(34,38);(3,7)(6,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,31)(18,34)(19,35)(20,25)(21,37)(23,39)(24,40)(26,30)(32,38)(33,36);(1,3,4,7)(2,6,5,8)(12,27)(17,34)(22,39)(25,35)(26,40)(28,29)(31,37)(33,38)Polycyclic presentation:
2^8|a^2=ef,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,d^c=de,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29860, #6435583, #6435590, … |
| maximal normal subgroup of | C2×#29860, #6435583, #6435590, … |
| maximal quotient of | #330371, #330376, #330381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22848);