Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
440 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 107 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)(3,7,8,5)(9,10)(11,19)(12,15)(13,17)(14,18)(16,20)(21,23)(22,24);(1,3,4,8)(2,5,6,7)(9,10)(11,16)(12,17)(13,15)(14,18)(19,20)(21,24)(22,23);(10,15)(12,13)(16,19)(17,18)(20,22)(23,24);(2,6)(3,8)(9,11,14,21)(10,16,18,24)(12,22,13,20)(15,19,17,23)Polycyclic presentation:
2^8|a^2=e,b^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #151347, #151349, #151469, … |
| maximal normal subgroup of | #151347, #151349, #151469, … |
| maximal quotient of | #335080, #335087, #335088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23540);