Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| 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 | C2×2−1+4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
985 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,13,15)(11,19,21,32)(12,17,23,29)(14,26,25,18)(16,27,28,22)(20,33,34,40)(24,37,36,31)(30,38,39,35);(1,3,4,8)(2,5,6,7)(10,15)(16,28)(17,29)(18,26)(19,32)(30,39)(31,37)(33,40);(9,11,13,21)(10,16,15,28)(12,20,23,34)(14,27,25,22)(17,30,29,39)(18,32,26,19)(24,38,36,35)(31,40,37,33);(9,12)(10,17)(11,20)(13,23)(14,24)(15,29)(16,30)(18,31)(19,33)(21,34)(22,35)(25,36)(26,37)(27,38)(28,39)(32,40);(12,23)(17,29)(20,34)(24,36)(30,39)(31,37)(33,40)(35,38)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,g^c=gh,e^d=eh61 conjugacy classes — view the full character table.
| Fitting subgroup of | #60499⋊C3 |
| maximal subgroup of | #6574580, #6574583, #6574585, … |
| maximal normal subgroup of | #6574580, #6574583, #6574585, … |
| maximal quotient of | #6370841, #6371349, #6371351, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53487);