Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| 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 | C2×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
442 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 442 conjugacy classes of subgroups — too many to draw. Only its 127 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)(9,10,14,18)(11,20,16,23)(12,21,22,15)(13,19,24,17);(9,11)(10,12)(13,21)(14,16)(15,24)(17,23)(18,22)(19,20);(1,3,4,8)(2,5,6,7)(9,12,14,22)(10,15,18,21)(11,17,16,19)(13,20,24,23);(9,12,14,22)(10,16,18,11)(13,23,24,20)(15,19,21,17)Polycyclic presentation:
2^8|a^2=f,c^2=f,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,g^c=gh,e^d=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | #151189, #151191, #151384, … |
| maximal normal subgroup of | #151189, #151191, #151384, … |
| maximal quotient of | #305242, #305260, #305263, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18971);