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 |
| prime radical | #3487 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1015 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(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);(1,3)(2,5)(4,8)(6,7)(9,11,15,23)(10,16,20,31)(12,21,26,37)(13,27,14,22)(17,29,32,40)(18,33,19,30)(24,35,34,39)(25,38,28,36);(9,12,15,26)(10,17,20,32)(11,21,23,37)(13,25,14,28)(16,29,31,40)(18,24,19,34)(22,36,27,38)(30,39,33,35);(3,8)(5,7)(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);(2,6)(5,7)(11,23)(16,31)(21,37)(22,27)(29,40)(30,33)(35,39)(36,38)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,d^2=h,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 | #152138, #244527, #244531, … |
| maximal normal subgroup of | #152138, #244527, #244531, … |
| maximal quotient of | #6386050, #6387655, #6387675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54640);