Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator, prime radical | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
285 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 285 conjugacy classes of subgroups — too many to draw. Only its 83 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:
(2,4)(5,6)(7,11,12,9)(8,10)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(1,2,3,4)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(5,7,8,12)(6,9,10,11)(15,21)(18,24)(22,27)(25,28)Polycyclic presentation:
2^8|a^2=f,b^2=d,c^2=g,f^2=h,b^a=bd,c^a=ce,e^a=eg,c^b=ch,f^c=fg52 conjugacy classes — view the full character table.
| maximal subgroup of | #97767, #97769, #97771, … |
| maximal normal subgroup of | #97767, #97769, #97771, … |
| maximal quotient of | #98801, #98803, #98805, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1550);