Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| 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 | #532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
882 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)(3,7)(4,6)(5,8)(9,10,15,19)(11,16,22,32)(12,23,26,37)(13,18,14,28)(17,24,33,25)(20,34,36,39)(21,30,27,31)(29,38,40,35);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,27)(14,21)(15,22)(17,29)(18,31)(19,32)(23,34)(24,38)(25,35)(26,36)(28,30)(33,40)(37,39);(9,12,13,24,15,26,14,25)(10,17,28,37,19,33,18,23)(11,20,27,38,22,36,21,35)(16,29,30,39,32,40,31,34);(3,8)(5,7)(9,11)(10,16)(12,20)(13,27)(14,21)(15,22)(17,29)(18,31)(19,32)(23,34)(24,38)(25,35)(26,36)(28,30)(33,40)(37,39);(2,6)(5,7)(9,13,15,14)(10,18,19,28)(11,21,22,27)(12,24,26,25)(16,30,32,31)(17,23,33,37)(20,35,36,38)(29,39,40,34)Polycyclic presentation:
2^8|a^2=h,c^2=gh,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130235, #130238, #130436, … |
| maximal normal subgroup of | #130235, #130238, #130436, … |
| maximal quotient of | #6386764, #6389170, #6389316, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55317);