Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,7)(5,6)(10,13)(11,12)(18,20)(19,23)(25,26,30,27)(28,31,35,33)(29,32,37,34)(36,38,40,39);(1,3)(2,4)(5,8)(6,7)(9,10,14,13)(11,16,12,15)(17,18,21,20)(19,22,23,24)(26,27)(28,35)(32,34)(36,40);(2,5)(3,7)(9,11,14,12)(10,15,13,16)(17,19)(18,22)(20,24)(21,23)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39);(1,4,6,8)(2,7,5,3)(9,12,14,11)(10,15,13,16)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(1,5,6,2)(3,8,7,4)(9,13,14,10)(11,15,12,16)(17,20,21,18)(19,22,23,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=fg,b^2=fhi,c^2=fgi,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2168024);