Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,36)(28,38)(29,37)(30,39)(31,41)(32,40)(33,43)(34,44)(35,42)(45,47)(46,48);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,29)(18,23,25,37)(20,30,32,46)(21,33,35,28)(24,39,40,48)(26,43,42,38)(27,34,45,31)(36,44,47,41);(3,7)(6,8)(10,14)(13,16)(17,20,22,32)(18,24,25,40)(19,27,29,45)(21,34,35,31)(23,36,37,47)(26,44,42,41)(28,46,33,30)(38,48,43,39);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(18,25)(23,37)(24,40)(26,42)(36,47)(38,43)(39,48)(41,44)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,g^b=gh,d^c=de,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #123036, #123046, #142974, … |
| maximal normal subgroup of | #123036, #123046, #142974, … |
| maximal quotient of | #391199, #391270, #391415, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25756);