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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
910 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)(17,26)(20,34)(21,30)(22,31)(23,32)(25,33)(27,38)(29,36)(35,40)(37,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,29)(18,30)(19,32)(24,34)(25,35)(26,36)(27,37)(28,31)(33,40)(38,39);(9,12,13,25,15,27,14,26)(10,17,28,38,19,33,18,24)(11,20,21,35,23,37,22,36)(16,29,31,39,32,40,30,34);(3,8)(5,7)(9,13,15,14)(10,18,19,28)(11,21,23,22)(12,25,27,26)(16,30,32,31)(17,24,33,38)(20,35,37,36)(29,34,40,39);(2,6)(5,7)(9,13,15,14)(10,18,19,28)(11,22,23,21)(12,25,27,26)(16,31,32,30)(17,24,33,38)(20,36,37,35)(29,39,40,34)Polycyclic presentation:
2^8|c^2=gh,d^2=h,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130285, #130287, #486337, … |
| maximal normal subgroup of | #130285, #130287, #486337, … |
| maximal quotient of | #6386751, #6389139, #6389163, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55263);