Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10530 |
| automorphism | Aut(G) = C23×D43 (order 4096) · Inn(G) = #394 · Out(G) = C26 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
182 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 49 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:
(1,2,7,12)(3,13,14,16)(4,8,15,9)(5,11,6,10)(17,18,21,25)(19,26,27,30)(20,22,28,24)(23,31,32,29);(1,3)(2,8)(4,6)(5,15)(7,14)(9,12)(10,16)(11,13)(18,22)(19,27)(20,28)(23,30)(24,25)(26,32);(1,4,5,14,7,15,6,3)(2,9,10,16,12,8,11,13)(17,19,28,31,21,27,20,29)(18,23,22,30,25,32,24,26)Polycyclic presentation:
2^8|a^2=g,c^2=dh,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fgh,e^c=egh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10533, #238987, #238988, … |
| maximal normal subgroup of | C2×#10533, #238987, #238988, … |
| maximal quotient of | #104880, #104885, #104887, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3521);