Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
333 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 333 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,26)(20,30)(21,25)(22,24)(23,29)(27,32)(28,31);(2,5)(6,8)(17,19)(18,23)(20,28)(21,29)(22,27)(24,31)(25,26)(30,32);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,27)(20,22)(23,31)(24,30)(26,32)(28,29);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,24,25,30)(19,28,29,27)(23,31,26,32)Polycyclic presentation:
2^8|a^2=eh,c^2=eh,d^2=fh,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gh,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #150364, #150366, #150704, … |
| maximal normal subgroup of | #150364, #150366, #150704, … |
| maximal quotient of | #380477, #380479, #381207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23078);