Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10304 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
335 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 335 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,11)(10,12)(13,14,18,22)(15,23,26,27)(16,21,17,20)(19,25,28,24);(2,4)(5,6)(7,9)(8,11)(10,12)(13,15,18,26)(14,19,22,28)(16,25,17,24)(20,27,21,23);(6,9)(7,12)(10,11)(15,24)(16,17)(19,23)(20,21)(25,26)(27,28)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,c^b=cf,d^b=dg,e^b=eh,f^b=fgh,d^c=dg,e^c=eh,f^c=fgh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94643, #125649, #125723, … |
| maximal normal subgroup of | #94643, #125649, #125723, … |
| maximal quotient of | #102788, #102794, #102845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3294);