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 | #3515 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
878 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)(6,8)(9,10,15,19)(11,16,23,33)(12,25,28,36)(13,20,14,18)(17,29,35,27)(21,37,26,40)(22,32,24,31)(30,39,34,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,26)(13,22)(14,24)(15,23)(17,34)(18,31)(19,33)(20,32)(21,28)(25,40)(27,39)(29,38)(30,35)(36,37);(9,12)(10,17)(11,21)(13,27)(14,29)(15,28)(16,30)(18,25)(19,35)(20,36)(22,38)(23,26)(24,39)(31,37)(32,40)(33,34);(2,5)(6,8)(9,13,15,14)(10,18,19,20)(11,22,23,24)(12,27,28,29)(16,31,33,32)(17,25,35,36)(21,38,26,39)(30,37,34,40);(1,3,4,7)(2,6,5,8)(10,19)(11,23)(17,35)(18,20)(21,26)(22,24)(25,36)(38,39)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=eh,g^a=gh,c^b=ch,d^b=df,e^b=eh,g^c=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | #6358979, #6358983, #6359011, … |
| maximal normal subgroup of | #6358979, #6358983, #6359011, … |
| maximal quotient of | #6379600, #6379605, #6380406, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54540);