Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#447 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
850 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 850 conjugacy classes of subgroups — too many to draw. Only its 159 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:
(2,3)(9,10)(11,17)(12,14)(13,21)(15,19)(16,20)(18,23)(22,24);(1,2,4,3)(5,6)(7,8)(10,14)(15,22)(16,21)(17,23);(6,8)(9,11,13,19,12,18,20,24)(10,15,16,17,14,22,21,23);(1,3,4,2)(10,14)(15,22)(16,21)(17,23)Polycyclic presentation:
2^8|b^2=eh,c^2=f,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg58 conjugacy classes — view the full character table.
| maximal subgroup of | #294283, #294284, #294303, … |
| maximal normal subgroup of | #294283, #294284, #294303, … |
| maximal quotient of | #297218, #297226, #297234, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18712);