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) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
339 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18,20,23)(19,25,27,22)(21,24,29,31)(26,30,28,32);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,16,15)(17,19,20,27)(18,22,23,25)(21,26,29,28)(24,32,31,30);(1,3,5,4)(2,7,8,6)(9,12,13,11)(10,15,16,14)(18,23)(19,26)(21,29)(22,30)(25,32)(27,28);(1,4,5,3)(2,7,8,6)(10,16)(14,15)(18,23)(22,25)(24,31)(30,32)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=eh,d^2=ef,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27104, #6446824, #6446843, … |
| maximal normal subgroup of | C2×#27104, #6446824, #6446843, … |
| maximal quotient of | #361235, #362342, #365927, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20092);