Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
764 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 764 conjugacy classes of subgroups — too many to draw. Only its 137 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,18,21)(12,15,19,22)(17,23,24,20);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,17)(13,18)(15,20)(16,21)(19,24)(22,23);(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,24)(21,23)Polycyclic presentation:
2^8|a^2=h,c^2=e,b^a=be,c^a=cf,d^a=dg,g^c=gh,f^d=fh49 conjugacy classes — view the full character table.
| maximal subgroup of | #245236, #245237, #245238, … |
| maximal normal subgroup of | #245236, #245237, #245238, … |
| maximal quotient of | #349824, #349825, #349826, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26010);