Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10378 |
| automorphism | Aut(G) = C23×#35346 (order 2048) · Inn(G) = #393 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #393 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #120 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
176 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 45 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:
(1,2,4,7)(3,10)(5,9,6,14)(8,11)(12,15)(13,16)(17,18)(19,25)(20,28)(21,24)(22,26)(23,29)(27,32)(30,31);(2,7)(3,11,13,12)(8,16,15,10)(9,14)(20,29)(22,31)(23,28)(26,30);(1,3)(2,8)(4,13)(5,11)(6,12)(7,15)(9,16)(10,14)(17,19,21,27)(18,22,24,31)(20,30,29,26)(23,32,28,25)Polycyclic presentation:
2^8|a^2=d,b^2=dg,c^2=h,e^2=gh,f^2=g,b^a=bd,c^a=ce,e^a=eh,f^a=fgh,c^b=cf,e^b=eh,d^c=dg,e^c=egh,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10381, #238314, #238315, … |
| maximal normal subgroup of | C2×#10381, #238314, #238315, … |
| maximal quotient of | #104734, #104735, #104738, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3369);