Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle, prime radical | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
305 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 305 conjugacy classes of subgroups — too many to draw. Only its 159 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,8)(5,7)(9,10)(11,13)(12,15)(14,16)(17,18,23,19,22,26,21,25)(20,24,28,27,29,31,30,32);(1,3,5,4)(2,6,7,8)(9,11,13,10)(12,15,16,14)(17,19,23,26,22,25,21,18)(20,27,28,31,29,32,30,24);(1,4,5,3)(2,6,7,8)(10,11)(14,15)(17,20,22,29)(18,24,26,31)(19,27,25,32)(21,30,23,28);(9,12,13,16)(10,14,11,15)(17,21,22,23)(18,25,26,19)(20,28,29,30)(24,27,31,32)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=ef,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=ceh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #244479, #244480, #244481, … |
| maximal normal subgroup of | #244479, #244480, #244481, … |
| maximal quotient of | #355237, #355389, #355863, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15300);