Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1333 conjugacy classes of subgroups; only the 991 normal subgroups are shown.
This group has 1333 conjugacy classes of subgroups — too many to draw. Only its 991 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,13,10,14,20,26)(7,11,17,23,16,22,29,33)(8,12,19,25,18,24,30,34)(15,21,28,32,27,31,35,36);(1,2,3,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,21)(13,23)(14,22)(18,27)(19,28)(20,29)(24,31)(25,32)(26,33)(30,35)(34,36);(5,8)(6,12)(7,15)(9,19)(10,18)(11,21)(13,25)(14,24)(16,27)(17,28)(20,30)(22,31)(23,32)(26,34)(29,35)(33,36);(8,18)(12,24)(15,27)(19,30)(21,31)(25,34)(28,35)(32,36);(7,16)(11,22)(15,27)(17,29)(21,31)(23,33)(28,35)(32,36)Polycyclic presentation:
2^8|a^2=g,b^2=f,g^2=h,b^a=bf,e^b=eh,d^c=dh88 conjugacy classes — view the full character table.
| Fitting subgroup of | #60100⋊C3 |
| maximal subgroup of | #129243, #129244, #129265, … |
| maximal normal subgroup of | #129243, #129244, #129265, … |
| maximal quotient of | #6344427, #6344428, #6344429, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53088);