Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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:
(3,7)(6,8)(9,10)(11,16)(12,20)(13,21)(14,15)(17,18)(19,23)(22,24)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2)(3,6)(4,5)(7,8)(10,15)(12,21)(16,23)(17,22)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(2,5)(6,8)(9,11,14,19)(10,16,15,23)(12,22,21,17)(13,18,20,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,13)(11,17)(14,21)(15,20)(16,24)(18,23)(19,22)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,3)(2,6)(4,7)(5,8)(9,13)(10,12)(11,18)(14,20)(15,21)(16,22)(17,23)(19,24)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=f,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#52709, #8092148, #8092476, … |
| maximal normal subgroup of | C2×#52709, #8092148, #8092476, … |
| maximal quotient of | #582307, #582523, #582556, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45697);