Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25440 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
309 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 309 conjugacy classes of subgroups — too many to draw. Only its 143 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:
(1,2,4,6)(3,7,8,5)(9,10)(11,17)(12,24)(13,16)(14,19)(15,20)(18,25)(21,29)(22,36)(23,31)(26,34)(27,30)(28,33)(32,35)(37,39)(38,40);(10,16,20,19)(11,21,23,28)(17,31)(18,30,34,24)(22,35,37,38)(29,33)(32,40)(36,39);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,22)(13,21)(14,28)(15,23)(16,29)(18,32)(19,33)(20,31)(24,36)(25,35)(26,38)(27,37)(30,39)(34,40);(9,12,13,25,15,27,14,26)(10,18,19,24,20,34,16,30)(11,22,21,35,23,37,28,38)(17,32,33,36,31,40,29,39)Polycyclic presentation:
2^8|a^2=f,b^2=eg,c^2=f,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #144514, #144526, #145865, … |
| maximal normal subgroup of | #144514, #144526, #145865, … |
| maximal quotient of | #325797, #325799, #325800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21811);