Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24278 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
370 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 123 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:
(1,2,4,6)(3,5,7,8)(10,15)(11,18)(12,22)(13,17)(20,21)(23,24);(1,3)(2,5)(4,7)(6,8)(9,10,13,15,14,12,17,22)(11,16,20,23,21,19,18,24);(3,7)(5,8)(9,11)(10,16)(12,19)(13,20)(14,21)(15,23)(17,18)(22,24);(9,12,13,22,14,10,17,15)(11,19,20,24,21,16,18,23)Polycyclic presentation:
2^8|a^2=g,b^2=e,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #150474, #301958, #301959, … |
| maximal normal subgroup of | #150474, #301958, #301959, … |
| maximal quotient of | #314653, #314658, #314664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20660);