Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = C23×#6342371 (order 4096) · Inn(G) = C25 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | #3529 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,17,14,18,25,32)(11,15,21,28,19,26,33,37)(12,16,24,31,23,30,36,40)(20,27,34,38,22,29,35,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,22)(13,21)(14,19)(16,29)(17,28)(18,26)(20,23)(24,35)(25,33)(27,30)(31,39)(32,37)(34,36)(38,40);(3,8)(5,7)(11,19)(12,23)(15,26)(16,30)(21,33)(24,36)(28,37)(31,40);(2,6)(5,7)(11,19)(15,26)(20,22)(21,33)(27,29)(28,37)(34,35)(38,39);(9,12,13,24,14,23,25,36)(10,16,17,31,18,30,32,40)(11,20,21,34,19,22,33,35)(15,27,28,38,26,29,37,39)Polycyclic presentation:
2^8|a^2=g,e^2=g,g^2=h,b^a=bf,d^a=df,c^b=cfh,d^b=dh,e^b=eh,e^c=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #129664, #210396, #210397, … |
| maximal normal subgroup of | #129664, #210396, #210397, … |
| maximal quotient of | #6370422, #6370484, #6370485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53315);