Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18414 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3561 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
482 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 482 conjugacy classes of subgroups — too many to draw. Only its 113 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 — 4 generators on 40 points:
(2,6)(3,4)(7,8)(9,10,14,18)(11,19,22,32)(12,16,23,30)(13,25,27,17)(15,26,29,21)(20,33,34,40)(24,37,36,31)(28,38,39,35);(1,2,5,7)(3,6,4,8)(9,11,14,22)(10,15,18,29)(12,20,23,34)(13,26,27,21)(16,28,30,39)(17,32,25,19)(24,38,36,35)(31,40,37,33);(3,4)(6,8)(9,12)(10,16)(11,20)(13,24)(14,23)(15,28)(17,31)(18,30)(19,33)(21,35)(22,34)(25,37)(26,38)(27,36)(29,39)(32,40);(1,3,5,4)(2,6,7,8)(12,23)(16,30)(20,34)(24,36)(28,39)(31,37)(33,40)(35,38)Polycyclic presentation:
2^8|a^2=gh,b^2=g,d^2=h,e^2=g,b^a=be,c^a=cf,d^a=dh,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #250396, #250398, #250399, … |
| maximal normal subgroup of | #250396, #250398, #250399, … |
| maximal quotient of | #169655, #169669, #169720, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11427);