Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
754 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)(9,10,13,16)(11,18,19,15)(12,20,21,24)(14,23,22,17);(1,2,4,3)(5,6)(7,8)(9,11,13,19)(10,15,16,18)(12,17,21,23)(14,20,22,24);(6,8)(9,12,14,15,13,21,22,18)(10,11,17,20,16,19,23,24);(1,3,4,2)(10,16)(11,19)(17,23)(20,24)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=f,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25737, #6443094, #6443104, … |
| maximal normal subgroup of | C2×#25737, #6443094, #6443104, … |
| maximal quotient of | #297320, #297321, #297332, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18725);