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)(11,20)(12,22)(13,18)(14,24)(15,19)(16,17)(21,23);(1,3,5,8)(2,6,7,4)(9,11,13,17)(10,16,18,20)(12,19,23,24)(14,21,15,22);(9,12,13,23)(10,17,18,11)(14,20,15,16)(19,21,24,22);(1,4,5,6)(2,3,7,8)(9,11)(10,12)(13,17)(14,22)(15,21)(16,24)(18,23)(19,20)Polycyclic presentation:
2^8|a^2=eh,b^2=e,c^2=h,d^2=eh,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 | #151304, #151306, #151442, … |
| maximal normal subgroup of | #151304, #151306, #151442, … |
| maximal quotient of | #341466, #341467, #341470, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23971);