Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13605 |
| automorphism | Aut(G) = C2×#493397 (order 1024) · Inn(G) = #2127 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2127 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | #124 | C23 |
| prime radical | #535 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #535 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
166 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 43 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11,13,26)(4,15,16,31)(5,18,20,9)(6,10)(7,12,22,29)(8,17)(14,24,19,27)(21,28,23,30)(25,32);(1,3,6,14)(2,7,10,21)(4,12,16,28)(5,19,20,13)(8,24,25,11)(9,22,18,23)(15,27,31,26)(17,29,32,30);(4,16)(7,21)(8,25)(12,28)(15,31)(17,32)(22,23)(29,30)Polycyclic presentation:
2^8|a^2=e,b^2=h,d^2=g,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=cg,d^b=dg,e^b=efgh,f^b=fh,d^c=dh,e^d=egh,f^e=fh25 conjugacy classes — view the full character table.
| maximal subgroup of | #152134, #152135, #152136, … |
| maximal normal subgroup of | #152134, #152135, #152136, … |
| maximal quotient of | #149594, #149635, #149713, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6595);