Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13697 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
87 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10,17,24)(11,19,29,49)(12,18,32,46)(13,21,35,42)(14,20,25,51)(15,38,40,67)(16,41,39,23)(22,54,55,33)(26,47,56,71)(27,60,62,72)(28,63,61,48)(30,43,64,68)(31,65,58,45)(34,66,36,53)(37,52,57,50)(44,69,70,59);(2,5)(6,8)(9,11,25,56,15,27,34,31,39,61,37,59,54,64,13,12,17,29,14,26,40,62,36,58,16,28,57,69,33,30,35,32)(10,18,42,68,22,44,50,48,41,65,53,60,67,71,20,19,24,46,21,43,55,70,52,63,23,45,66,72,38,47,51,49);(1,3,4,7)(2,6,5,8)(9,12,13,30,33,59,37,28,16,31,34,62,40,56,25,29,17,32,35,64,54,69,57,61,39,58,36,27,15,26,14,11)(10,19,20,47,38,60,53,45,23,48,50,70,55,68,42,46,24,49,51,71,67,72,66,65,41,63,52,44,22,43,21,18)Polycyclic presentation:
2^8|a^2=h,b^2=eh,c^2=d,d^2=fg,e^2=f,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=efg,f^a=fg,g^a=gh,c^b=ch46 conjugacy classes — view the full character table.
| maximal subgroup of | #151552, #151553, #151564, … |
| maximal normal subgroup of | #151552, #151553, #151564, … |
| maximal quotient of | #152386, #152833, #152893, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6712);