Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
497 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 497 conjugacy classes of subgroups — too many to draw. Only its 119 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) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,13)(8,16,17,11)(9,12)(10,15,18,20)(14,19);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,13)(12,18)(16,19)(17,20);(7,14)(10,18)(13,19)(15,20)Polycyclic presentation:
2^8|a^2=e,b^2=f,b^a=bd,d^a=dg,g^a=gh,c^b=ch,e^b=eg,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #146475, #146476, #146497, … |
| maximal normal subgroup of | #146475, #146476, #146497, … |
| maximal quotient of | #126909, #126948, #126973, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5562);