Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
537 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,19)(17,27)(18,22)(20,25)(21,24)(26,28);(1,2)(3,4)(6,9)(10,12)(13,15,17,16,18,25,24,26)(14,19,21,23,22,28,27,20);(2,4)(5,7,8,11)(6,10,9,12)(13,16,24,15,18,26,17,25)(14,20,27,28,22,23,21,19);(6,9)(10,12)(13,15,24,26,18,25,17,16)(14,19,27,20,22,28,21,23)Polycyclic presentation:
2^8|b^2=f,c^2=eh,d^2=fh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27281, #6440757, #6440764, … |
| maximal normal subgroup of | C2×#27281, #6440757, #6440764, … |
| maximal quotient of | #314222, #314224, #314230, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20269);