Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#546 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
837 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 837 conjugacy classes of subgroups — too many to draw. Only its 153 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(6,8)(9,10)(11,19)(12,15)(13,18)(14,17)(16,22)(20,23)(21,24);(1,2,4,3)(10,15)(11,20)(13,14)(16,19)(21,22)(23,24);(5,6,7,8)(9,11,13,22,12,21,14,20)(10,16,17,19,15,23,18,24);(1,3,4,2)(10,15)(11,21)(17,18)(20,22)Polycyclic presentation:
2^8|b^2=eh,c^2=f,d^2=eh,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=gh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139228, #139296, #268052, … |
| maximal normal subgroup of | #139228, #139296, #268052, … |
| maximal quotient of | #297413, #297422, #297427, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18757);