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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1005 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)(11,16)(12,18)(13,19)(14,28)(15,20)(17,24)(21,30)(22,31)(23,33)(25,34)(26,32)(27,35)(29,37)(36,39)(38,40);(2,5)(6,8)(9,11,15,23)(10,16,20,33)(12,21,25,38)(13,26,14,22)(17,29,35,39)(18,30,34,40)(19,32,28,31)(24,37,27,36);(9,12)(10,17)(11,21)(13,24)(14,27)(15,25)(16,29)(18,28)(19,34)(20,35)(22,36)(23,38)(26,37)(30,31)(32,40)(33,39);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,21)(13,27)(14,24)(15,25)(16,30)(17,28)(19,35)(20,34)(22,37)(23,38)(26,36)(29,31)(32,39)(33,40);(9,13,15,14)(10,19,20,28)(11,22,23,26)(12,24,25,27)(16,31,33,32)(17,34,35,18)(21,36,38,37)(29,30,39,40)Polycyclic presentation:
2^8|b^2=h,d^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,e^b=eh,g^c=gh,e^d=eh61 conjugacy classes — view the full character table.
| maximal subgroup of | #6575438, #6575445, #6575450, … |
| maximal normal subgroup of | #6575438, #6575445, #6575450, … |
| maximal quotient of | #6370993, #6371815, #6371816, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53586);