Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | Aut(G) order 65536 |
| 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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
442 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 442 conjugacy classes of subgroups — too many to draw. Only its 137 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)(13,14,16,17)(15,19,20,18);(5,6,8,9)(7,10,11,12)(14,17)(15,18)(19,20);(2,4)(5,7)(6,10)(8,11)(9,12)(13,15,16,20)(14,18,17,19);(7,11)(10,12)(15,20)(18,19)Polycyclic presentation:
2^8|a^2=f,b^2=eh,c^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 | #100646, #100647, #103272, … |
| maximal normal subgroup of | #100646, #100647, #103272, … |
| maximal quotient of | #329791, #329792, #329793, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22745);