Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C23 |
| commutator, socle, prime radical | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
401 conjugacy classes of subgroups; only the 203 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 203 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) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,18,21,17,20,26,16)(15,22,25,24,23,19,27,28);(5,7,9,8)(6,10,11,12)(13,15,17,23)(14,19,20,22)(16,24,21,28)(18,25,26,27);(1,2,3,4)(15,23)(19,22)(24,28)(25,27);(5,8,9,7)(6,10,11,12)(13,16,26,20,17,21,18,14)(15,24,27,22,23,28,25,19)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=f,d^2=eg,g^2=h,b^a=be,c^a=cf,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | #122417, #122425, #124920, … |
| maximal normal subgroup of | #122417, #122425, #124920, … |
| maximal quotient of | #213426, #213427, #213428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14703);