Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = #72285904 (order 1024) · Inn(G) = C22⋊C4 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | C22⋊C4 |
| commutator | C22 | C4×C16 |
| Frattini, cosocle | C22×C16 | C22 |
| socle | C2 | #1374 |
| prime radical | C23 | C2×C16 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
87 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,9,5,11,18,29,6,12,19,30,21,33,42,53,7,10,20,28,22,31,43,51,23,32,44,52,45,54,60,63)(3,13,15,34,16,35,39,55,17,36,40,56,41,57,59,64,14,8,24,25,37,26,46,48,38,27,47,49,58,50,61,62);(1,3)(2,8,10,13)(4,15)(5,16)(6,17)(7,14)(9,25,28,34)(11,26,31,35)(12,27,32,36)(18,39)(19,40)(20,24)(21,41)(22,37)(23,38)(29,48,51,55)(30,49,52,56)(33,50,54,57)(42,59)(43,46)(44,47)(45,58)(53,62,63,64)(60,61)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=f,f^2=g,g^2=h,b^a=bc,e^a=eh,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7469, #151883, #151895, … |
| maximal normal subgroup of | C2×#7469, #151883, #151895, … |
| maximal quotient of | #93993, #94000, #94007, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 457);