Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#454 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
676 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 147 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(7,13)(9,17)(11,19)(12,16)(14,20)(15,18);(1,2,3,4)(5,6,10,8)(7,11,15,14)(9,12,17,16)(13,19,18,20);(5,7)(6,11)(8,14)(9,18)(10,15)(12,20)(13,17)(16,19);(5,8,10,6)(7,14,15,11)(9,12,17,16)(13,19,18,20)Polycyclic presentation:
2^8|b^2=eh,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,f^c=fh,g^c=gh,f^d=fh49 conjugacy classes — view the full character table.
| maximal subgroup of | #500515, #500516, #500517, … |
| maximal normal subgroup of | #500515, #500516, #500517, … |
| maximal quotient of | #347520, #347521, #347522, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25913);