Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60454 |
| 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 | #3060 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
921 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)(3,6)(4,5)(7,8)(9,10,14,18)(11,15,20,28)(12,22,24,37)(13,26,23,17)(16,30,31,25)(19,32,34,40)(21,36,33,29)(27,38,39,35);(2,5)(6,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,27)(17,29)(18,28)(22,32)(23,33)(24,34)(25,35)(26,36)(30,38)(31,39)(37,40);(9,12,23,30,14,24,13,25)(10,16,26,37,18,31,17,22)(11,19,33,38,20,34,21,35)(15,27,36,40,28,39,29,32);(1,3,4,7)(2,6,5,8)(12,24)(16,31)(19,34)(22,37)(25,30)(27,39)(32,40)(35,38);(11,20)(15,28)(19,34)(21,33)(27,39)(29,36)(32,40)(35,38)Polycyclic presentation:
2^8|a^2=h,c^2=gh,d^2=f,g^2=h,b^a=bf,c^a=cg,g^a=gh,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #482710, #495574, #499270, … |
| maximal normal subgroup of | #482710, #495574, #499270, … |
| maximal quotient of | #6371185, #6372119, #6372121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53687);