Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23906 |
| 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 | C22×C4 | C24 |
| prime radical | #3485 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
657 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 657 conjugacy classes of subgroups — too many to draw. Only its 127 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)(11,21)(12,16)(13,20)(14,19)(15,18)(17,23)(22,24);(2,5)(6,8)(9,11,14,12)(10,16,19,21)(13,22,15,23)(17,18,24,20);(1,3,4,7)(2,6,5,8)(9,12,14,11)(10,17,19,24)(13,23,15,22)(16,20,21,18);(2,5)(6,8)(9,13,14,15)(10,18,19,20)(11,22,12,23)(16,24,21,17)Polycyclic presentation:
2^8|b^2=h,c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139769, #139820, #139821, … |
| maximal normal subgroup of | #139769, #139820, #139821, … |
| maximal quotient of | #254570, #254571, #254578, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17009);