Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13077 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1893 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3058 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
243 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 73 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:
(1,2,7,10,6,12,23,11)(3,13,9,22,16,27,15,5)(4,8,19,25,18,26,20,29)(14,31,24,17,32,30,28,21);(1,3,5,2,7,15,13,11,6,16,22,12,23,9,27,10)(4,14,21,8,20,24,30,25,18,32,17,26,19,28,31,29);(1,4)(2,8)(3,14)(5,21)(6,18)(7,20)(9,28)(10,29)(11,25)(12,26)(13,30)(15,24)(16,32)(17,22)(19,23)(27,31)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,d^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #130234, #130236, #130284, … |
| maximal normal subgroup of | #130234, #130236, #130284, … |
| maximal quotient of | #137622, #137662, #137987, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6067);