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) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
303 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 303 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7)(3,4,8,6)(9,10,14,19)(11,15,22,23)(12,20,16,21)(13,18,24,17);(1,3,5,8)(2,6,7,4)(9,11)(10,15)(12,17)(13,20)(14,22)(16,18)(19,23)(21,24);(9,12,14,16)(10,11,19,22)(13,15,24,23)(17,21,18,20);(1,4,5,6)(2,3,7,8)(9,11)(10,16)(12,19)(13,21)(14,22)(15,18)(17,23)(20,24)Polycyclic presentation:
2^8|a^2=eh,b^2=e,c^2=h,d^2=e,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 | #151305, #151307, #151443, … |
| maximal normal subgroup of | #151305, #151307, #151443, … |
| maximal quotient of | #341465, #341468, #341469, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23970);