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 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
485 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,9,12)(7,11,8,10)(13,14,17,21)(15,20,23,19)(16,25,24,28)(18,27,26,22);(5,7,9,8)(6,10,12,11)(13,15,17,23)(14,19,21,20)(16,26,24,18)(22,25,27,28);(1,2,3,4)(13,16,18,15,17,24,26,23)(14,20,22,25,21,19,27,28);(5,8,9,7)(6,11,12,10)(14,21)(19,20)(22,27)(25,28)Polycyclic presentation:
2^8|a^2=e,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×#25842, #6445729, #6445740, … |
| maximal normal subgroup of | C2×#25842, #6445729, #6445740, … |
| maximal quotient of | #302173, #302183, #302194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18830);