Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9471 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
229 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 229 conjugacy classes of subgroups — too many to draw. Only its 59 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 — 3 generators on 32 points:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(17,18,21,25,22,19,23,27)(20,26,29,32,28,24,30,31);(1,3)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16)(17,19,21,27,22,18,23,25)(20,26,30,31,28,24,29,32);(3,11)(7,15)(10,16)(12,13)(17,20)(18,24)(19,26)(21,30)(22,28)(23,29)(25,32)(27,31)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,e^b=eh,d^c=dh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132887, #132888, #132891, … |
| maximal normal subgroup of | #132887, #132888, #132891, … |
| maximal quotient of | #100163, #100164, #100169, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2481);