Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
361 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 361 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:
(1,2,5,8)(3,4,7,6)(9,10,15,17,13,19,14,20)(11,21,22,32,24,16,25,33)(12,26,30,18,28,36,29,35)(23,38,39,34,27,40,37,31);(1,3,5,7)(2,6,8,4)(9,11)(10,16)(12,27)(13,24)(14,25)(15,22)(17,33)(18,34)(19,21)(20,32)(23,28)(26,38)(29,39)(30,37)(31,35)(36,40);(1,4,5,6)(2,7,8,3)(10,17)(11,22)(14,15)(16,21)(18,26)(19,20)(23,37)(24,25)(27,39)(29,30)(31,34)(35,36);(2,8)(3,7)(9,12)(10,18)(11,23)(13,28)(14,30)(15,29)(16,31)(17,26)(19,35)(20,36)(21,34)(22,37)(24,27)(25,39)(32,38)(33,40)Polycyclic presentation:
2^8|a^2=ef,b^2=eh,c^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #125729, #126214, #126388, … |
| maximal normal subgroup of | #125729, #126214, #126388, … |
| maximal quotient of | #345115, #345118, #345123, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24221);