Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
693 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,4)(6,8)(9,10,14,18)(11,19,23,33)(12,16,24,30)(13,26,20,17)(15,27,29,22)(21,32,35,40)(25,37,34,31)(28,38,39,36);(9,11,20,27,14,23,13,22)(10,15,26,33,18,29,17,19)(12,21,34,38,24,35,25,36)(16,28,37,40,30,39,31,32);(1,2,3,4)(9,12)(10,16)(11,21)(13,25)(14,24)(15,28)(17,31)(18,30)(19,32)(20,34)(22,36)(23,35)(26,37)(27,38)(29,39)(33,40);(5,6,7,8)(12,24)(16,30)(21,35)(25,34)(28,39)(31,37)(32,40)(36,38)Polycyclic presentation:
2^8|a^2=h,b^2=eh,c^2=f,d^2=g,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,d^c=dh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #33309⋊C3 |
| maximal subgroup of | #141096, #144086, #144088, … |
| maximal normal subgroup of | #141096, #144086, #144088, … |
| maximal quotient of | #352908, #352912, #352914, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26297);