Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
489 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 489 conjugacy classes of subgroups — too many to draw. Only its 187 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,20,24,21,23,27,30)(19,22,25,29,26,28,31,32);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,21,26)(18,22,23,28)(20,25,27,31)(24,29,30,32);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,23)(20,27)(22,28)(25,31);(2,8)(6,9)(7,16)(14,15)Polycyclic presentation:
2^8|a^2=g,b^2=eh,c^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #107189, #108273, #110102, … |
| maximal normal subgroup of | #107189, #108273, #110102, … |
| maximal quotient of | #215595, #215600, #215603, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14818);