Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3488 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
882 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 882 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,10,14)(7,15,17,20)(8,13,9,12)(11,18,19,16);(1,2)(3,4)(5,7)(6,11)(8,18)(9,16)(10,17)(12,20)(13,15)(14,19);(2,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,15)(13,20)(14,19);(5,7)(6,11)(8,18)(9,16)(10,17)(12,20)(13,15)(14,19)Polycyclic presentation:
2^8|a^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,e^b=eh,f^c=fh,e^d=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | #273315, #273317, #273351, … |
| maximal normal subgroup of | #273315, #273317, #273351, … |
| maximal quotient of | #244656, #244666, #244671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16865);