Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
538 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 538 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(9,10)(11,18)(12,20)(13,15)(14,23)(16,19)(17,22)(21,24);(1,2)(3,4)(5,6)(7,8)(10,15)(16,24)(17,23)(18,20);(6,8)(9,11,13,12)(10,16,15,24)(14,21,22,19)(17,20,23,18);(2,4)(9,12,13,11)(10,16,15,24)(14,19,22,21)(17,20,23,18)Polycyclic presentation:
2^8|a^2=eh,c^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=deh,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #142524, #142531, #143064, … |
| maximal normal subgroup of | #142524, #142531, #143064, … |
| maximal quotient of | #332969, #332970, #332971, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23505);