Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2856 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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 — 4 generators on 20 points:
(1,2,3,4)(6,9)(10,12)(14,18)(19,20);(5,6,8,9)(7,10,11,12)(13,14,16,18)(15,20,17,19);(2,4)(5,7)(6,10)(8,11)(9,12)(14,19)(15,17)(18,20);(7,11)(10,12)(13,15,16,17)(14,20,18,19)Polycyclic presentation:
2^8|a^2=f,b^2=e,d^2=h,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,d^c=de,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #100696, #100698, #168515, … |
| maximal normal subgroup of | #100696, #100698, #168515, … |
| maximal quotient of | #329795, #329796, #329797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22746);