Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator | C23 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
465 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 465 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) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,8,13,9,14,20,28,10,12,21,25,22,26,33,35)(7,15,17,27,18,29,31,36,16,11,19,23,30,24,32,34);(5,7)(6,11)(8,19)(9,18)(10,16)(12,15)(13,27)(14,24)(17,21)(20,32)(22,30)(23,25)(26,29)(28,36)(31,33)(34,35);(2,4)(7,16)(11,15)(17,19)(18,30)(23,27)(24,29)(31,32)(34,36)Polycyclic presentation:
2^8|a^2=f,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=ch,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93480, #93481, #93486, … |
| maximal normal subgroup of | #93480, #93481, #93486, … |
| maximal quotient of | #129202, #129207, #129241, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5670);