Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| 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 | C2×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
866 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)(6,7)(9,10,14,18)(11,20,22,16)(12,24,26,34)(13,19,15,28)(17,29,33,25)(21,36,37,40)(23,32,27,31)(30,39,35,38);(1,2,4,5)(3,7,8,6)(9,11,14,22)(10,16,18,20)(12,21,26,37)(13,27,15,23)(17,30,33,35)(19,32,28,31)(24,40,34,36)(25,39,29,38);(9,12)(10,17)(11,21)(13,25)(14,26)(15,29)(16,30)(18,33)(19,34)(20,35)(22,37)(23,38)(24,28)(27,39)(31,40)(32,36);(1,3,4,8)(2,6,5,7)(10,18)(11,22)(17,33)(19,28)(21,37)(23,27)(24,34)(38,39);(9,13,14,15)(10,19,18,28)(11,23,22,27)(12,25,26,29)(16,31,20,32)(17,34,33,24)(21,38,37,39)(30,40,35,36)Polycyclic presentation:
2^8|a^2=h,b^2=f,d^2=fh,e^2=h,g^2=h,b^a=bf,c^a=cg,d^a=dh,g^a=gh,d^b=df,e^b=eh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6576436, #6576437, #6576440, … |
| maximal normal subgroup of | #6576436, #6576437, #6576440, … |
| maximal quotient of | #6379397, #6379468, #6380336, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54326);