Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10382 |
| 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 | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
198 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 198 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)(3,9)(5,8,13,15)(6,14)(10,11)(12,16)(17,18)(19,22)(20,27,29,23)(21,24)(25,30,28,32)(26,31);(1,2)(3,10,12,6)(4,7)(5,8)(9,11,16,14)(13,15)(17,19,21,26)(18,22,24,31)(20,28)(23,32)(25,29)(27,30);(1,3)(2,6)(4,12)(5,14)(7,10)(8,16)(9,15)(11,13)(17,20,21,29)(18,23,24,27)(19,25,26,28)(22,30,31,32)Polycyclic presentation:
2^8|a^2=d,b^2=dg,c^2=h,e^2=gh,b^a=bd,c^a=ce,e^a=eh,f^a=fgh,c^b=cf,e^b=eh,f^b=fg,d^c=dg,e^c=egh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10385, #238374, #238375, … |
| maximal normal subgroup of | C2×#10385, #238374, #238375, … |
| maximal quotient of | #104758, #104759, #104762, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3373);