Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| 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 | #3499 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1007 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,15,19)(11,16,22,32)(12,23,25,33)(13,18,14,27)(17,28,34,24)(20,35,38,40)(21,30,26,31)(29,36,39,37);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,26)(14,21)(15,22)(17,29)(18,31)(19,32)(23,35)(24,37)(25,38)(27,30)(28,36)(33,40)(34,39);(9,12,15,25)(10,17,19,34)(11,20,22,38)(13,24,14,28)(16,29,32,39)(18,33,27,23)(21,36,26,37)(30,35,31,40);(3,8)(5,7)(9,11)(10,16)(12,20)(13,26)(14,21)(15,22)(17,29)(18,31)(19,32)(23,35)(24,37)(25,38)(27,30)(28,36)(33,40)(34,39);(9,13,15,14)(10,18,19,27)(11,21,22,26)(12,24,25,28)(16,30,32,31)(17,33,34,23)(20,36,38,37)(29,35,39,40)Polycyclic presentation:
2^8|a^2=fh,c^2=h,e^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,d^b=df,e^b=eh,g^c=gh,e^d=eh55 conjugacy classes — view the full character table.
| maximal subgroup of | #250089, #250121, #250153, … |
| maximal normal subgroup of | #250089, #250121, #250153, … |
| maximal quotient of | #6379094, #6379319, #6379737, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54198);