Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #380 |
| prime radical | C24 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
231 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 231 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,19)(12,23,25,16)(13,26)(14,18)(15,21)(17,27)(20,33,35,31)(22,32)(24,30,39,38)(28,34,37,40)(29,36);(9,11,13,21,14,22,27,36)(10,15,17,19,18,29,26,32)(12,20,24,34,25,35,39,40)(16,28,30,31,23,37,38,33);(9,12)(10,16)(11,20)(13,24)(14,25)(15,28)(17,30)(18,23)(19,31)(21,34)(22,35)(26,38)(27,39)(29,37)(32,33)(36,40)Polycyclic presentation:
2^8|a^2=f,b^2=d,d^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^c=fg64 conjugacy classes — view the full character table.
| maximal subgroup of | #93325, #93326, #93575, … |
| maximal normal subgroup of | #93325, #93326, #93575, … |
| maximal quotient of | #98816, #98817, #98818, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1589);