Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| 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 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
899 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,16)(12,24)(13,18)(14,28)(15,19)(17,27)(20,33)(21,30)(22,31)(23,37)(25,32)(26,38)(29,35)(34,39)(36,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,29)(18,30)(19,31)(24,33)(25,34)(26,36)(27,35)(28,37)(32,39)(38,40);(9,12,14,27,15,26,13,25)(10,17,18,24,19,32,28,38)(11,20,23,35,22,36,21,34)(16,29,30,33,31,39,37,40);(3,8)(5,7)(9,13,15,14)(10,18,19,28)(11,21,22,23)(12,25,26,27)(16,30,31,37)(17,24,32,38)(20,34,36,35)(29,33,39,40);(2,6)(5,7)(11,22)(12,26)(16,31)(17,32)(21,23)(24,38)(25,27)(30,37)Polycyclic presentation:
2^8|c^2=g,d^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,e^c=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#62309, #10577613, #10577659, … |
| maximal normal subgroup of | C2×#62309, #10577613, #10577659, … |
| maximal quotient of | #6386798, #6389160, #6389220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55297);