Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| 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 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1059 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)(9,10,14,18)(11,19,20,32)(12,16,23,30)(13,25,27,17)(15,26,28,22)(21,33,34,40)(24,37,36,31)(29,38,39,35);(1,3)(2,5)(4,8)(6,7)(11,20)(15,28)(19,32)(21,34)(22,26)(29,39)(33,40)(35,38);(3,8)(5,7)(9,11,14,20)(10,15,18,28)(12,21,23,34)(13,26,27,22)(16,29,30,39)(17,32,25,19)(24,38,36,35)(31,40,37,33);(2,6)(5,7)(9,12)(10,16)(11,21)(13,24)(14,23)(15,29)(17,31)(18,30)(19,33)(20,34)(22,35)(25,37)(26,38)(27,36)(28,39)(32,40);(12,23)(16,30)(21,34)(24,36)(29,39)(31,37)(33,40)(35,38)Polycyclic presentation:
2^8|a^2=h,c^2=h,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cfh,g^c=gh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #131351, #131353, #131355, … |
| maximal normal subgroup of | #131351, #131353, #131355, … |
| maximal quotient of | #6374226, #6374871, #6375541, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53761);