Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
905 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)(3,8)(4,6)(5,7)(9,10,13,18,14,17,24,30)(11,15,21,27,20,26,33,37)(12,16,23,29,22,28,34,38)(19,25,31,36,32,35,39,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,25)(17,26)(18,27)(22,32)(23,31)(24,33)(28,35)(29,36)(30,37)(34,39)(38,40);(1,4,5,3)(2,6,7,8)(9,12,13,23,14,22,24,34)(10,16,18,29,17,28,30,38)(11,19,21,31,20,32,33,39)(15,25,27,36,26,35,37,40);(11,20)(12,22)(15,26)(16,28)(21,33)(23,34)(27,37)(29,38);(10,17)(15,26)(16,28)(18,30)(25,35)(27,37)(29,38)(36,40)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=fg,g^2=h,b^a=bf,e^a=eh,d^b=dh,d^c=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #129387, #129388, #129392, … |
| maximal normal subgroup of | #129387, #129388, #129392, … |
| maximal quotient of | #6369900, #6369901, #6370119, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53181);