Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | Aut(G) = C2×#72293743 (order 2048) · Inn(G) = #386 · Out(G) = C2×C22⋊C4 |
| 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 | #3213 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
219 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 219 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(1,2,5,8,6,10,4,9)(3,11,13,15,14,16,12,7)(17,18,21,25,22,24,31,19)(20,27,29,32,28,23,30,26);(1,3)(2,7,10,15)(4,12)(5,13)(6,14)(8,11,9,16)(17,19,21,24,22,25,31,18)(20,26,30,27,28,32,29,23);(2,8)(4,5)(7,11)(9,10)(12,13)(15,16)(17,20)(18,23)(19,26)(21,30)(22,28)(24,27)(25,32)(29,31)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=gh,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=egh,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9333, #199568, #199569, … |
| maximal normal subgroup of | C2×#9333, #199568, #199569, … |
| maximal quotient of | #99995, #100001, #100003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2321);