Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| 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 | |
325 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 167 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)(17,18,22,25,21,24,26,19)(20,28,30,27,29,23,31,32);(1,3,5,4)(2,6,7,8)(9,11,13,10)(12,15,16,14)(17,19,26,24,21,25,22,18)(20,27,31,28,29,32,30,23);(1,4,5,3)(2,6,7,8)(10,11)(14,15)(17,20,21,29)(18,23,24,28)(19,27,25,32)(22,30,26,31);(9,12,13,16)(10,14,11,15)(20,29)(23,28)(27,32)(30,31)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=de,c^b=ce,d^c=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #244473, #244474, #244475, … |
| maximal normal subgroup of | #244473, #244474, #244475, … |
| maximal quotient of | #355233, #355383, #355855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15296);