Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
473 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(9,10,15,23,14,19,13,18)(11,16,20,24,12,22,21,17);(1,2,4,3)(5,6)(7,8)(9,11)(10,16)(12,14)(13,21)(15,20)(17,18)(19,22)(23,24);(6,8)(9,12,14,11)(10,17,19,24)(13,21,15,20)(16,23,22,18);(1,3,4,2)(9,13,14,15)(10,18,19,23)(11,20,12,21)(16,24,22,17)Polycyclic presentation:
2^8|a^2=fh,b^2=e,c^2=h,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24769, #6442983, #6442990, … |
| maximal normal subgroup of | C2×#24769, #6442983, #6442990, … |
| maximal quotient of | #278704, #278706, #278712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17757);