Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8626 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #380 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #117 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
160 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 65 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) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,8,4,6,12,14)(3,9,11,16,10,7,13,15)(17,18)(19,28)(20,33,35,40)(21,36)(22,26)(23,27)(24,30)(25,34,41,37)(29,38,45,48)(31,39)(32,43)(42,44,47,46);(2,6)(7,9)(8,14)(15,16)(17,19,21,30,23,32,22,31)(18,24,26,28,27,39,36,43)(20,29,34,44,35,45,37,46)(25,38,40,42,41,48,33,47);(1,3,4,10)(2,7,6,9)(5,13,12,11)(8,16,14,15)(17,20)(18,25)(19,29)(21,34)(22,37)(23,35)(24,38)(26,40)(27,41)(28,42)(30,44)(31,46)(32,45)(33,36)(39,48)(43,47)Polycyclic presentation:
2^8|a^2=f,b^2=dh,c^2=h,d^2=g,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,e^c=eg,f^c=fgh40 conjugacy classes — view the full character table.
| maximal subgroup of | #130472, #130473, #131074, … |
| maximal normal subgroup of | #130472, #130473, #131074, … |
| maximal quotient of | #98841, #98843, #98853, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1623);