Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10488 |
| 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 | |
183 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 183 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:
(2,6)(3,9)(4,12)(7,15)(8,13)(10,14)(17,18)(19,22)(20,23)(21,25)(24,29)(26,30)(27,32)(28,31);(1,2,5,8)(3,10,9,7)(4,13,12,6)(11,14,16,15)(17,19,21,27)(18,22,25,32)(20,28,29,26)(23,31,24,30);(1,3,4,11,5,9,12,16)(2,7,6,14,8,10,13,15)(18,23)(20,29)(22,30)(24,25)(26,28)(31,32)Polycyclic presentation:
2^8|b^2=gh,c^2=d,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,c^b=cf,d^b=dg,e^b=eh,e^c=egh,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10491, #240556, #240559, … |
| maximal normal subgroup of | C2×#10491, #240556, #240559, … |
| maximal quotient of | #104907, #104908, #104913, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3479);