Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24802 |
| automorphism | Aut(G) order 32768 |
| 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 | C23 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
315 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 315 conjugacy classes of subgroups — too many to draw. Only its 139 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10,14,17)(11,18,20,15)(12,21,16,13)(19,23,24,22);(1,3,5,4)(2,6,8,7)(9,11,13,19,14,20,21,24)(10,15,12,22,17,18,16,23);(9,12,13,10,14,16,21,17)(11,15,19,23,20,18,24,22);(1,4,5,3)(2,7,8,6)(9,13,14,21)(10,16,17,12)(11,19,20,24)(15,23,18,22)Polycyclic presentation:
2^8|a^2=e,b^2=egh,c^2=f,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #263837, #263838, #263877, … |
| maximal normal subgroup of | #263837, #263838, #263877, … |
| maximal quotient of | #325118, #325123, #325127, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21688);