Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
678 conjugacy classes of subgroups; only the 407 normal subgroups are shown.
This group has 678 conjugacy classes of subgroups — too many to draw. Only its 407 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) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(5,8)(12,19)(16,21)(18,23)(22,24)(25,26,30,31)(27,32,33,28)(29,37,38,34)(35,36,40,39);(1,2,4,6)(3,5,7,8)(10,14)(11,17)(16,21)(18,23)(26,31)(28,32)(34,37)(36,39);(1,3,4,7)(2,5,6,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,30,38)(26,34,31,37)(27,35,33,40)(28,36,32,39)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=fgh,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133248, #133253, #251745, … |
| maximal normal subgroup of | #133248, #133253, #251745, … |
| maximal quotient of | #553765, #553797, #553997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30421);