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) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
784 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 784 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) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,15)(12,17)(13,16)(14,18)(19,24)(20,22)(21,23);(2,4)(9,11,13,21)(10,15,16,23)(12,20,14,19)(17,22,18,24);(6,8)(10,16)(11,19)(12,14)(15,22)(20,21)(23,24);(2,4)(9,12,13,14)(10,17,16,18)(11,20,21,19)(15,22,23,24)Polycyclic presentation:
2^8|a^2=fh,b^2=h,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #273379, #273396, #273415, … |
| maximal normal subgroup of | #273379, #273396, #273415, … |
| maximal quotient of | #267181, #267182, #267183, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17361);