Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
501 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 501 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) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,9,12)(7,11,8,10)(13,14)(15,19)(16,24)(17,26)(18,21)(20,25)(22,28)(23,27);(5,7,9,8)(6,10,12,11)(13,15,18,23)(14,19,21,27)(16,25,22,17)(20,28,26,24);(1,2,3,4)(5,7,9,8)(6,11,12,10)(13,16,17,15,18,22,25,23)(14,19,20,24,21,27,26,28);(5,8,9,7)(6,11,12,10)Polycyclic presentation:
2^8|a^2=e,b^2=eh,c^2=ef,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #138781, #250916, #250996, … |
| maximal normal subgroup of | #138781, #250916, #250996, … |
| maximal quotient of | #304311, #304322, #304329, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18919);