Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11858 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
157 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 157 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,12,14)(13,16)(17,18)(19,29)(20,26)(21,36)(22,38)(23,28)(24,31)(25,34)(27,37)(30,46)(32,40)(33,43)(35,45)(39,44)(41,48)(42,47);(1,3)(2,6)(4,12)(5,11)(7,15)(8,9)(10,13)(14,16)(17,19,23,33)(18,24,28,42)(20,31,35,47)(21,30,34,48)(22,39,37,32)(25,43,36,29)(26,44,45,40)(27,46,38,41);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,22,34,23,35,37,21)(18,25,27,26,28,36,38,45)(19,30,32,47,33,48,39,31)(24,40,41,29,42,44,46,43)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=f,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,c^b=cfh,d^b=dg,e^b=efgh,f^b=fh,g^b=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #131727, #131728, #131729, … |
| maximal normal subgroup of | #131727, #131728, #131729, … |
| maximal quotient of | #124302, #124455, #124475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4847);