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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
653 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,7,8)(3,4,5,6)(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,26,30,31)(27,32,35,38)(28,33,37,40)(29,36,39,34);(1,3,7,5)(2,4,8,6)(10,14)(15,21)(16,19)(22,23)(26,31)(27,35)(28,37)(29,39);(1,4,7,6)(2,3,8,5)(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,27)(26,32)(28,34)(29,33)(30,35)(31,38)(36,37)(39,40);(1,5,7,3)(2,4,8,6)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,28)(26,33)(27,36)(29,38)(30,37)(31,40)(32,39)(34,35);(1,6,7,4)(2,3,8,5)(9,12)(10,16)(11,17)(13,20)(14,19)(15,22)(18,24)(21,23)(25,29,30,39)(26,34,31,36)(27,33,35,40)(28,38,37,32)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=fgh,d^2=g,e^2=gh,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×#58976, #8105184, #8105196, … |
| maximal normal subgroup of | C2×#58976, #8105184, #8105196, … |
| maximal quotient of | #586117, #586128, #586143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51964);