Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| 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, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,15)(11,16,12,14)(17,18)(19,22)(20,24)(21,23)(25,29)(26,30)(27,31)(28,32);(9,11,13,12)(10,14,15,16)(19,25)(20,27)(22,29)(24,31)(26,28)(30,32);(2,5)(6,8)(11,12)(14,16)(17,19,21,26)(18,22,23,30)(20,28,27,25)(24,32,31,29);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(18,23)(22,30)(24,31)(29,32)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=h,d^2=ef,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gh,d^c=de,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#30159, #6434840, #6434846, … |
| maximal normal subgroup of | C2×#30159, #6434840, #6434846, … |
| maximal quotient of | #380450, #380455, #380500, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23147);