Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| 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×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
791 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)(6,8)(9,10)(11,18)(12,20)(13,16)(14,23)(15,19)(17,22)(21,24);(1,2,4,3)(9,11,13,19)(10,15,16,18)(12,14,21,22)(17,24,23,20);(5,6,7,8)(9,12,14,19,13,21,22,11)(10,15,17,20,16,18,23,24);(1,3,4,2)(10,16)(15,18)(17,23)(20,24)Polycyclic presentation:
2^8|b^2=e,c^2=f,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25761, #6443145, #6443156, … |
| maximal normal subgroup of | C2×#25761, #6443145, #6443156, … |
| maximal quotient of | #297359, #297369, #297380, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18749);