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) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2407 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,13,18,14,19,15,20)(11,17,23,24,21,16,12,22);(2,4)(5,6)(7,8)(9,11,15,12,14,21,13,23)(10,16,20,24,19,17,18,22);(6,8)(9,12)(10,17)(11,15)(13,21)(14,23)(16,19)(18,22)(20,24);(2,4)(9,13,14,15)(10,18,19,20)(11,12,21,23)(16,24,17,22)Polycyclic presentation:
2^8|a^2=fh,b^2=f,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25137, #6438942, #6438951, … |
| maximal normal subgroup of | C2×#25137, #6438942, #6438951, … |
| maximal quotient of | #282332, #282334, #282340, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18125);