Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(10,14)(15,18)(16,21)(22,24)(25,26,30,28)(27,32,35,31)(29,36,38,33)(34,39,37,40);(10,14)(11,17)(12,20)(22,24)(25,27,30,35)(26,31,28,32)(29,37,38,34)(33,40,36,39);(1,3,4,8)(2,5,6,7)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,27)(26,32)(28,31)(29,34)(30,35)(33,40)(36,39)(37,38);(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,28,30,26)(27,32,35,31)(29,36,38,33)(34,40,37,39);(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29,30,38)(26,33,28,36)(27,34,35,37)(31,39,32,40)Polycyclic presentation:
2^8|a^2=gh,b^2=f,c^2=g,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#51795, #7707827, #7707840, … |
| maximal normal subgroup of | C2×#51795, #7707827, #7707840, … |
| maximal quotient of | #560593, #560594, #561703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44783);