Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
583 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,5)(6,8)(9,10,12,13)(11,14,15,16)(17,18)(19,20)(21,22)(23,24)(26,27)(30,31);(3,7)(6,8)(10,13)(11,15)(18,22)(20,24)(25,26,29,27)(28,30,32,31);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,20)(21,23)(22,24)(25,27,29,26)(28,30,32,31);(1,3,4,7)(2,6,5,8)(9,10)(11,16)(12,13)(14,15)(17,20)(18,23)(19,22)(21,24)(25,28,29,32)(26,30,27,31)Polycyclic presentation:
2^8|a^2=g,b^2=eh,c^2=f,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#17270, #845126, #845134, … |
| maximal normal subgroup of | C2×#17270, #845126, #845134, … |
| maximal quotient of | #156545, #156547, #156548, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10258);