Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
670 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)(11,14)(12,15)(13,18)(16,21)(17,19)(20,23)(22,24)(25,26,30,28)(27,33,34,31)(29,32,37,36)(35,39,40,38);(1,2,5,4)(3,6,7,8)(10,11)(12,19)(13,21)(22,23)(27,34)(29,37)(31,33)(32,36);(1,3)(2,6)(4,8)(5,7)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(1,2,5,4)(3,6,7,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28,30,26)(27,31,34,33)(29,36,37,32)(35,38,40,39);(1,4,5,2)(3,8,7,6)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,37)(26,32,28,36)(27,35,34,40)(31,38,33,39)Polycyclic presentation:
2^8|a^2=fh,b^2=gh,c^2=f,d^2=g,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#47433, #7701358, #7701402, … |
| maximal normal subgroup of | C2×#47433, #7701358, #7701402, … |
| maximal quotient of | #576924, #577013, #577145, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40421);