Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×2−1+4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
499 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 499 conjugacy classes of subgroups — too many to draw. Only its 167 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:
(1,2,4,3)(6,11)(8,9)(12,20)(13,18)(14,15)(16,19);(1,3,4,2)(5,6)(7,14)(8,18)(9,11)(10,13)(12,16)(15,19)(17,20);(5,6)(7,15)(8,11)(9,18)(10,13)(12,17)(14,19)(16,20);(5,7)(6,12)(8,19)(9,16)(10,17)(11,20)(13,15)(14,18)Polycyclic presentation:
2^8|a^2=g,b^2=g,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=de,e^b=eh,d^c=dh,f^c=fh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #152465, #152466, #152558, … |
| maximal normal subgroup of | #152465, #152466, #152558, … |
| maximal quotient of | #226938, #226946, #226954, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15882);