Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| 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 | |
888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10)(11,21)(12,26)(13,20)(14,19)(15,18)(16,24)(17,27)(22,36)(23,31)(25,32)(28,34)(29,33)(30,39)(35,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,30)(18,31)(19,21)(20,32)(26,40)(27,37)(28,38)(29,39)(33,35)(34,36);(9,12,14,28)(10,17,19,33)(11,22,24,38)(13,27,15,29)(16,30,21,35)(18,26,20,34)(23,37,25,39)(31,40,32,36);(2,5)(6,8)(9,13,14,15)(10,18,19,20)(11,23,24,25)(12,27,28,29)(16,31,21,32)(17,26,33,34)(22,37,38,39)(30,40,35,36);(1,3,4,7)(2,6,5,8)(10,19)(11,24)(17,33)(18,20)(22,38)(23,25)(26,34)(37,39)Polycyclic presentation:
2^8|c^2=h,d^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gh,d^b=dfh,e^b=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #250350, #250351, #250382, … |
| maximal normal subgroup of | #250350, #250351, #250382, … |
| maximal quotient of | #6386472, #6388494, #6388612, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55079);