Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle, prime radical | C23 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
246 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 97 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) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(6,9)(7,11)(10,12)(13,14,18,19,17,16,15,20);(1,2,3,4)(13,15,17,18)(14,19,16,20);(5,6,8,10)(7,12,11,9)(13,16,17,14)(15,19,18,20)Polycyclic presentation:
2^8|a^2=f,b^2=d,c^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,e^a=eg,c^b=ch,e^c=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #98008, #98009, #98012, … |
| maximal normal subgroup of | #98008, #98009, #98012, … |
| maximal quotient of | #98801, #98802, #98810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1556);