Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#535 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
987 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 987 conjugacy classes of subgroups — too many to draw. Only its 239 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 — 4 generators on 20 points:
(2,4)(5,6,9,14,10,12,8,13)(7,15,18,11,17,20,16,19);(1,2,3,4)(5,7)(6,11)(8,18)(9,16)(10,17)(12,19)(13,20)(14,15);(6,12)(8,9)(11,19)(16,18);(6,12)(11,19)(13,14)(15,20)Polycyclic presentation:
2^8|a^2=g,b^2=eg,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #148483, #152322, #152324, … |
| maximal normal subgroup of | #148483, #152322, #152324, … |
| maximal quotient of | #193789, #193798, #193802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13941);