Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C25 | C23 |
| commutator | C23 | C2×C42 |
| prime radical | C26 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
2283 conjugacy classes of subgroups; only the 511 normal subgroups are shown.
This group has 2283 conjugacy classes of subgroups — too many to draw. Only its 511 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) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(6,8)(9,10,11,12)(13,14,15,16);(2,4)(6,8)(14,16)(17,18,19,20);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)Polycyclic presentation:
2^8|a^2=g,b^2=h,b^a=bd,c^a=ce,c^b=cf88 conjugacy classes — view the full character table.
| Fitting subgroup of | #7555⋊C3, #72240359, #72255747, … |
| maximal subgroup of | #93464, #93533, #97096, … |
| maximal normal subgroup of | #93464, #93533, #97096, … |
| maximal quotient of | #94849, #123507, #123508, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 543);