Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24464 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,10,7,11,13,9,15)(8,16,17,20,18,14,19,12);(1,2)(3,4)(5,7)(6,9)(8,14)(10,13)(11,15)(12,17)(16,18)(19,20);(2,4)(5,8)(6,12)(7,14)(9,17)(10,19)(11,18)(13,20)(15,16);(5,6,11,13)(7,9,15,10)(8,12,18,20)(14,17,16,19)Polycyclic presentation:
2^8|a^2=f,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,f^c=fh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24465, #6437925, #6437932, … |
| maximal normal subgroup of | C2×#24465, #6437925, #6437932, … |
| maximal quotient of | #268144, #268146, #268147, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17453);