Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×Q8 | C23 |
| prime radical | C2×#535 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
577 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 577 conjugacy classes of subgroups — too many to draw. Only its 75 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:
(1,2)(3,4)(5,6,9,14,10,15,7,12)(8,11,16,19,17,13,18,20);(6,11)(7,9)(8,17)(12,20)(13,15)(14,19);(2,4)(5,7,10,9)(6,12,15,14)(8,16,17,18)(11,19,13,20)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,c^b=ch,d^b=dh,f^b=fgh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94425, #94442, #133350, … |
| maximal normal subgroup of | #94425, #94442, #133350, … |
| maximal quotient of | #129219, #129252, #129340, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5691);