Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
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 | #2431 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
376 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 376 conjugacy classes of subgroups — too many to draw. Only its 63 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)(5,6,11,12,10,14,8,18)(7,15,13,20,19,9,17,16);(5,7,16,6,10,19,20,14)(8,17,15,12,11,13,9,18);(2,4)(5,8,10,11)(6,12,14,18)(7,17,19,13)(9,16,15,20)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=h,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dgh,f^b=fgh,g^b=gh,f^d=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #133374, #133377, #133379, … |
| maximal normal subgroup of | #133374, #133377, #133379, … |
| maximal quotient of | #137681, #137696, #137725, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6081);