Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
332 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(1,2,5,7)(3,4,8,6)(9,10,14,19,13,18,30,16)(11,20,21,33,23,15,24,32)(12,25,28,17,27,34,29,36)(22,38,39,35,26,40,37,31);(1,3,5,8)(2,6,7,4)(9,11,13,23)(10,15,18,20)(12,26,27,22)(14,21,30,24)(16,33,19,32)(17,35,36,31)(25,38,34,40)(28,37,29,39);(10,16)(11,21)(14,30)(17,34)(18,19)(22,37)(23,24)(25,36)(26,39)(28,29)(32,33)(38,40);(1,4,5,6)(2,3,7,8)(9,12,13,27)(10,17,18,36)(11,22,23,26)(14,29,30,28)(15,31,20,35)(16,34,19,25)(21,37,24,39)(32,40,33,38)Polycyclic presentation:
2^8|a^2=efh,b^2=e,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #153227, #6670990, #6670991, … |
| maximal normal subgroup of | #153227, #6670990, #6670991, … |
| maximal quotient of | #345521, #345522, #345526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24243);