Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11867 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1519 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C2×M16 | C23 |
| prime radical | #3058 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
196 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,11,7,12,6,22)(3,13,14,27,17,30,16,9)(4,8,18,10,21,25,20,24)(15,28,32,29,19,26,31,23);(2,8)(3,14)(4,18)(5,6)(9,26)(10,11)(12,25)(13,23)(16,17)(20,21)(22,24)(27,28)(29,30)(31,32);(1,3,6,16,7,17,5,14)(2,9,11,13,12,27,22,30)(4,19,18,31,21,15,20,32)(8,23,24,26,25,29,10,28)Polycyclic presentation:
2^8|a^2=e,c^2=g,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,g^a=gh,c^b=cf,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,d^c=dh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #130234, #130236, #130284, … |
| maximal normal subgroup of | #130234, #130236, #130284, … |
| maximal quotient of | #122774, #122775, #122776, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4858);