Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | Aut(G) order 65536 |
| 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 | C2×#535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
803 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 803 conjugacy classes of subgroups — too many to draw. Only its 175 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)(6,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,23)(19,24)(20,22);(5,6)(7,8)(9,11)(10,14)(12,18)(13,20)(15,23)(16,22)(17,24)(19,21);(2,4)(5,7,8,6)(10,15)(11,18)(12,21)(16,17)(19,20)(23,24);(10,16)(14,22)(15,17)(23,24)Polycyclic presentation:
2^8|c^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #394674, #394959, #395426, … |
| maximal normal subgroup of | #394674, #394959, #395426, … |
| maximal quotient of | #238471, #238477, #238481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16497);