Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
489 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 489 conjugacy classes of subgroups — too many to draw. Only its 155 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(6,8)(9,10,14,19,13,18,15,23)(11,16,21,17,20,24,12,22);(2,4)(9,11,15,21,13,20,14,12)(10,16,19,22,18,24,23,17);(5,6,7,8)(9,12,14,20,13,21,15,11)(10,17,23,24,18,22,19,16)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=efh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #97735, #97741, #97743, … |
| maximal normal subgroup of | #97735, #97741, #97743, … |
| maximal quotient of | #110077, #110079, #110081, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3951);