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 | |
889 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)(9,10,15,20)(11,16,23,31)(12,24,26,34)(13,19,14,18)(17,28,32,25)(21,35,37,39)(22,33,27,30)(29,36,40,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,21)(13,27)(14,22)(15,23)(17,29)(18,33)(19,30)(20,31)(24,35)(25,38)(26,37)(28,36)(32,40)(34,39);(9,12)(10,17)(11,21)(13,25)(14,28)(15,26)(16,29)(18,24)(19,34)(20,32)(22,36)(23,37)(27,38)(30,39)(31,40)(33,35);(2,5)(6,8)(9,13,15,14)(10,18,20,19)(11,22,23,27)(12,25,26,28)(16,30,31,33)(17,24,32,34)(21,36,37,38)(29,39,40,35);(1,3,4,7)(2,6,5,8)(11,23)(16,31)(21,37)(22,27)(29,40)(30,33)(35,39)(36,38)Polycyclic presentation:
2^8|a^2=h,d^2=h,e^2=f,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 | C2×#62080, #10578891, #10578916, … |
| maximal normal subgroup of | C2×#62080, #10578891, #10578916, … |
| maximal quotient of | #6386466, #6388486, #6388603, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55068);