Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62283 |
| 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 | #3551 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
890 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)(9,10,15,19)(11,16,22,33)(12,23,27,39)(13,18,14,29)(17,25,35,26)(20,36,24,38)(21,31,28,32)(30,40,34,37);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,24)(13,28)(14,21)(15,22)(17,34)(18,32)(19,33)(20,27)(23,38)(25,37)(26,40)(29,31)(30,35)(36,39);(9,12,14,26,15,27,13,25)(10,17,18,23,19,35,29,39)(11,20,21,37,22,24,28,40)(16,30,32,36,33,34,31,38);(3,8)(5,7)(9,11)(10,16)(12,20)(13,28)(14,21)(15,22)(17,30)(18,32)(19,33)(23,36)(24,27)(25,40)(26,37)(29,31)(34,35)(38,39);(2,6)(5,7)(9,13,15,14)(10,18,19,29)(11,21,22,28)(12,25,27,26)(16,31,33,32)(17,23,35,39)(20,37,24,40)(30,38,34,36)Polycyclic presentation:
2^8|a^2=h,c^2=g,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,c^b=ch,d^b=df,e^b=eh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #500812, #500819, #503234, … |
| maximal normal subgroup of | #500812, #500819, #503234, … |
| maximal quotient of | #6386767, #6389179, #6389181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55325);