Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#525 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,14,22,13,17,12,16)(11,18,19,15,21,23,20,24);(2,4)(5,6)(7,8);(6,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,24)(18,22);(2,4)(9,12,13,14)(10,16,17,22)(11,19,21,20)(15,23,24,18)Polycyclic presentation:
2^8|a^2=fh,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24383, #6437731, #6437750, … |
| maximal normal subgroup of | C2×#24383, #6437731, #6437750, … |
| maximal quotient of | #267056, #267062, #267068, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17371);