Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
693 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(10,14)(11,17)(16,21)(19,23)(25,26,29,30)(27,31,33,37)(28,32,35,38)(34,39,36,40);(1,2,5,3)(4,7,8,6)(12,20)(16,21)(19,23)(22,24)(26,30)(27,33)(28,35)(39,40);(2,3)(6,7)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,27)(26,31)(28,36)(29,33)(30,37)(32,40)(34,35)(38,39);(1,3,5,2)(4,7,8,6)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28)(26,32)(27,34)(29,35)(30,38)(31,39)(33,36)(37,40);(1,4)(2,6)(3,7)(5,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,39)(35,38)(36,40)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=g,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#58379, #8103990, #8104095, … |
| maximal normal subgroup of | C2×#58379, #8103990, #8104095, … |
| maximal quotient of | #586558, #586876, #587004, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51367);