Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| 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 | |
898 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)(9,10,14,19)(11,21,24,16)(12,25,27,34)(13,20,15,18)(17,29,32,26)(22,36,38,40)(23,31,28,33)(30,39,35,37);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,22)(13,28)(14,24)(15,23)(17,30)(18,33)(19,21)(20,31)(25,40)(26,39)(27,38)(29,37)(32,35)(34,36);(9,12)(10,17)(11,22)(13,26)(14,27)(15,29)(16,30)(18,25)(19,32)(20,34)(21,35)(23,37)(24,38)(28,39)(31,36)(33,40);(2,5)(6,8)(9,13,14,15)(10,18,19,20)(11,23,24,28)(12,26,27,29)(16,31,21,33)(17,25,32,34)(22,37,38,39)(30,36,35,40);(1,3,4,7)(2,6,5,8)(10,19)(11,24)(17,32)(18,20)(22,38)(23,28)(25,34)(37,39)Polycyclic presentation:
2^8|a^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=df,e^b=eh,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#62048, #10578845, #10578916, … |
| maximal normal subgroup of | C2×#62048, #10578845, #10578916, … |
| maximal quotient of | #6386448, #6388563, #6388616, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55036);