Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9914 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | #2617 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
428 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 428 conjugacy classes of subgroups — too many to draw. Only its 67 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 — 3 generators on 20 points:
(1,2)(3,4)(6,9)(7,11)(10,12)(13,14)(15,17)(16,19)(18,20);(5,6,8,10)(7,12,11,9)(14,17)(15,20)(18,19);(2,4)(15,20)(17,18)Polycyclic presentation:
2^8|b^2=gh,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fh,c^b=cf,d^b=dg,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9920, #234468, #234471, … |
| maximal normal subgroup of | C2×#9920, #234468, #234471, … |
| maximal quotient of | #102935, #102936, #102937, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2908);