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 | #2535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
524 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)(9,10,14,16,13,18,12,17)(11,15,19,24,20,22,21,23);(1,2,4,3)(5,6)(7,8)(9,11,12,21,13,20,14,19)(10,15,17,23,18,22,16,24);(6,8)(10,16)(11,19)(12,14)(15,22)(17,18)(20,21);(1,3,4,2)(11,20)(15,22)(19,21)(23,24)Polycyclic presentation:
2^8|a^2=fh,b^2=ef,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25400, #6444353, #6444359, … |
| maximal normal subgroup of | C2×#25400, #6444353, #6444359, … |
| maximal quotient of | #289467, #289469, #289475, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18388);