Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32935 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
481 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 481 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,19)(17,26)(18,20)(21,24)(22,27)(25,28);(6,9)(10,12)(13,15,17,16,18,24,27,25)(14,19,22,23,20,28,26,21);(1,2,3,4)(14,20)(19,28)(21,23)(22,26);(5,7,8,11)(6,10,9,12)(13,16,17,24,18,25,27,15)(14,21,22,19,20,23,26,28)Polycyclic presentation:
2^8|b^2=g,c^2=fh,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #103006, #103016, #104631, … |
| maximal normal subgroup of | #103006, #103016, #104631, … |
| maximal quotient of | #352640, #352646, #352647, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26221);