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 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10)(11,20)(12,24)(13,18)(14,19)(15,29)(16,23)(17,27)(21,35)(22,32)(25,33)(26,39)(28,31)(30,40)(34,36)(37,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,21)(13,28)(14,23)(15,22)(17,30)(18,32)(19,20)(24,38)(25,40)(26,37)(27,36)(29,31)(33,34)(35,39);(9,12,13,25,14,26,15,27)(10,17,29,39,19,33,18,24)(11,21,28,40,23,37,22,36)(16,30,31,35,20,34,32,38);(2,5)(6,8)(9,13,14,15)(10,18,19,29)(11,22,23,28)(12,25,26,27)(16,31,20,32)(17,24,33,39)(21,36,37,40)(30,35,34,38);(1,3,4,7)(2,6,5,8)(10,19)(11,23)(17,33)(18,29)(21,37)(22,28)(24,39)(36,40)Polycyclic presentation:
2^8|c^2=gh,d^2=h,e^2=fh,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 | #131663, #131665, #489374, … |
| maximal normal subgroup of | #131663, #131665, #489374, … |
| maximal quotient of | #6386891, #6389512, #6389540, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55427);