Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7654 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
239 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 239 conjugacy classes of subgroups — too many to draw. Only its 87 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) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6,9,10,8,11,7,12)(13,14)(15,21)(16,25)(17,19)(18,23)(20,22)(24,27)(26,28);(6,10)(7,9)(11,12)(13,15,22,28,17,24,16,23)(14,18,25,27,19,26,20,21);(1,2,3,4)(6,11)(10,12)(14,19)(18,26)(20,25)(21,27)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=egh,d^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dh,f^b=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #98474, #98475, #98476, … |
| maximal normal subgroup of | #98474, #98475, #98476, … |
| maximal quotient of | #101960, #101974, #102222, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2688);