Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
459 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 459 conjugacy classes of subgroups — too many to draw. Only its 139 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)(6,10)(7,11)(8,14)(12,16)(13,20)(15,19)(17,18);(5,6,11,18,8,13,15,16,9,12,19,20,14,17,7,10);(1,2,3,4);(6,12)(10,16)(13,17)(18,20)Polycyclic presentation:
2^8|b^2=eg,c^2=f,e^2=gh,g^2=h,b^a=be,c^a=cf,e^a=eg,g^a=gh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137545, #137546, #137549, … |
| maximal normal subgroup of | #137545, #137546, #137549, … |
| maximal quotient of | #491437, #491456, #491459, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26705);