Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
423 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 423 conjugacy classes of subgroups — too many to draw. Only its 107 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:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,23)(13,29)(15,22)(16,19)(17,27)(18,32)(20,28)(21,31)(24,26)(25,35)(30,34)(33,38)(36,40)(37,39);(1,3,4,8)(2,5,6,7)(9,11,14,10)(12,24,26,23)(13,18,30,21)(15,22,16,19)(17,28,20,27)(25,36,38,37)(29,31,34,32)(33,40,35,39);(9,12,14,26)(10,17,11,20)(13,25,30,38)(15,28,16,27)(18,33,21,35)(19,24,22,23)(29,37,34,36)(31,40,32,39);(2,6)(3,8)(9,13,14,30)(10,18,11,21)(12,25,26,38)(15,32,16,31)(17,33,20,35)(19,29,22,34)(23,36,24,37)(27,40,28,39)Polycyclic presentation:
2^8|a^2=eh,b^2=e,c^2=h,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #151351, #151353, #151465, … |
| maximal normal subgroup of | #151351, #151353, #151465, … |
| maximal quotient of | #335094, #335095, #335098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23544);