Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3062 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10)(11,19)(12,23)(13,17)(14,28)(15,21)(16,26)(18,27)(20,36)(22,31)(24,40)(25,34)(29,33)(30,39)(32,37)(35,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,24)(13,21)(14,22)(16,32)(17,19)(18,31)(20,25)(23,36)(26,39)(27,38)(28,35)(29,37)(30,33)(34,40);(2,5)(6,8)(9,12,13,25)(10,16,17,33)(11,20,21,24)(14,29,27,26)(15,30,19,32)(18,34,28,23)(22,39,38,37)(31,36,35,40);(1,3,4,7)(2,6,5,8)(10,17)(15,19)(16,33)(18,28)(23,34)(30,32)(31,35)(36,40);(11,21)(15,19)(20,24)(22,38)(30,32)(31,35)(36,40)(37,39)Polycyclic presentation:
2^8|c^2=h,d^2=fh,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cf,e^b=eh,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | #6670996, #6670999, #6671008, … |
| maximal normal subgroup of | #6670996, #6670999, #6671008, … |
| maximal quotient of | #6374524, #6374526, #6374937, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53915);