Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9415 |
| automorphism | Aut(G) = C2×#72293994 (order 4096) · Inn(G) = #386 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
212 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 212 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,6,8,4,9)(3,11,14,7,13,16,12,15)(17,18,22,26,23,25,21,19)(20,27,28,24,29,32,30,31);(1,3)(2,7)(4,14)(5,12)(6,13)(8,15)(9,16)(10,11)(17,19,22,25,23,26,21,18)(20,27,28,31,29,32,30,24);(2,8)(4,5)(7,15)(12,14)(17,20)(18,24)(19,27)(21,30)(22,28)(23,29)(25,31)(26,32)Polycyclic presentation:
2^8|a^2=f,b^2=df,d^2=gh,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=eg,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9419, #200765, #200774, … |
| maximal normal subgroup of | C2×#9419, #200765, #200774, … |
| maximal quotient of | #100091, #100092, #100101, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2407);