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 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
906 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)(11,18)(12,25)(13,21)(14,20)(15,19)(16,24)(17,29)(22,37)(23,32)(26,39)(27,35)(28,34)(30,36)(31,40)(33,38);(1,3)(2,5)(4,8)(6,7)(9,11,14,24)(10,16,20,18)(12,26,28,22)(13,30,15,23)(17,33,35,31)(19,36,21,32)(25,37,34,39)(27,38,29,40);(9,12,15,29,14,28,13,27)(10,17,21,25,20,35,19,34)(11,22,23,38,24,26,30,40)(16,31,32,39,18,33,36,37);(3,8)(5,7)(9,11)(10,18)(12,26)(13,30)(14,24)(15,23)(16,20)(17,31)(19,32)(21,36)(22,28)(25,39)(27,38)(29,40)(33,35)(34,37);(2,6)(5,7)(9,13,14,15)(10,19,20,21)(11,23,24,30)(12,27,28,29)(16,32,18,36)(17,34,35,25)(22,38,26,40)(31,39,33,37)Polycyclic presentation:
2^8|b^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,d^c=dh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131736, #131739, #489595, … |
| maximal normal subgroup of | #131736, #131739, #489595, … |
| maximal quotient of | #6386771, #6389184, #6389185, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55332);