Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C25 | C23 |
| prime radical | C26 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1531 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 1531 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,13,15,16)(12,14);(2,4)(9,11)(10,13,14,16)(12,15);(6,8)Polycyclic presentation:
2^8|a^2=f,b^2=g,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=dh,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #105752, #106076, #106654, … |
| maximal normal subgroup of | #105752, #106076, #106654, … |
| maximal quotient of | C22⋊#1304, #105705, #105707, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3723);