Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10326 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | #3192 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
349 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 349 conjugacy classes of subgroups — too many to draw. Only its 53 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 — 3 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,14)(12,15)(13,16)(17,22)(18,21)(19,23)(20,24);(2,5)(3,8)(6,7)(9,11)(10,14)(12,17)(13,18)(15,22)(16,21)(19,24)(20,23);(3,8)(5,6)(9,12,13,19)(10,15,16,23)(11,17,18,24)(14,20,21,22)Polycyclic presentation:
2^8|c^2=h,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fgh,c^b=cf,d^b=dg,e^b=eh,d^c=dg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10328, #234360, #234361, … |
| maximal normal subgroup of | C2×#10328, #234360, #234361, … |
| maximal quotient of | #102591, #102592, #102593, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3316);