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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(9,10)(11,13)(12,19)(14,18)(15,21)(16,22)(17,23)(20,24);(1,2,4,3)(5,6)(7,8)(9,11,13,10)(12,17,20,15)(14,18,22,16)(19,21,24,23);(6,8)(9,12,14,21,13,20,22,23)(10,15,16,19,11,17,18,24);(1,3,4,2)(10,11)(15,17)(16,18)(19,24)Polycyclic presentation:
2^8|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×#25736, #6443094, #6443103, … |
| maximal normal subgroup of | C2×#25736, #6443094, #6443103, … |
| maximal quotient of | #297319, #297322, #297331, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18724);