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 | #2006 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
376 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 376 conjugacy classes of subgroups — too many to draw. Only its 111 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)(3,6)(4,5)(7,8)(9,10)(11,19)(12,20)(13,18)(14,17)(15,22)(16,24)(21,23);(9,11)(10,12)(13,15)(14,21)(16,23)(17,22)(18,19)(20,24);(2,5)(6,8)(9,12)(10,15)(11,16)(13,19)(14,22)(17,20)(18,21)(23,24);(1,3,4,7)(2,6,5,8)(9,12)(10,11)(13,23)(14,22)(15,16)(17,21)(18,20)(19,24)Polycyclic presentation:
2^8|d^2=fh,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 | #151221, #151223, #151401, … |
| maximal normal subgroup of | #151221, #151223, #151401, … |
| maximal quotient of | #309450, #309468, #309471, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19567);