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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,8,4,6)(10,14)(11,17)(16,21)(19,23)(25,26,30,34)(27,32,28,31)(29,37,38,33)(35,39,36,40);(1,3)(2,6)(4,5)(7,8)(12,20)(16,21)(19,23)(22,24)(25,27,30,28)(26,31,34,32)(29,36,38,35)(33,40,37,39);(3,4)(6,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28,30,27)(26,32,34,31)(29,36,38,35)(33,40,37,39);(1,4,5,3)(2,6,7,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,26)(27,32)(28,31)(29,33)(30,34)(35,40)(36,39)(37,38);(1,2)(3,8)(4,6)(5,7)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29,30,38)(26,33,34,37)(27,35,28,36)(31,39,32,40)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=fh,d^2=h,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×#58756, #8104991, #8104997, … |
| maximal normal subgroup of | C2×#58756, #8104991, #8104997, … |
| maximal quotient of | #586147, #586765, #586973, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51744);