Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24283 |
| automorphism | Aut(G) order 8192 |
| 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 | #3003 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
494 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 494 conjugacy classes of subgroups — too many to draw. Only its 115 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)(3,7)(4,6)(5,8)(9,10)(11,19)(12,17)(13,16)(14,18)(15,21)(20,23)(22,24);(1,3,4,8)(2,5,6,7)(9,11,13,20,12,21,14,22)(10,15,16,24,17,19,18,23);(10,16)(11,20)(13,14)(15,19)(17,18)(21,22);(2,6)(5,7)(10,17)(11,21)(16,18)(20,22)Polycyclic presentation:
2^8|b^2=eg,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #149958, C2×#28137, #6459457, … |
| maximal normal subgroup of | #149958, C2×#28137, #6459457, … |
| maximal quotient of | #316367, #316370, #316379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21125);