Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(6,7)(9,10,14,12)(11,16,13,15)(17,18,21,19)(20,23,24,22)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39);(2,4)(6,7)(9,10,14,12)(11,15,13,16)(26,31)(27,33)(32,39)(34,40);(1,2,5,4)(3,7,8,6)(9,11,14,13)(10,15,12,16)(17,19,21,18)(20,23,24,22)(25,27,30,33)(26,28,31,36)(29,34,38,40)(32,35,39,37);(1,3,5,8)(2,6,4,7)(9,12,14,10)(11,15,13,16)(17,20)(18,22)(19,23)(21,24)(25,28)(26,27)(29,37)(30,36)(31,33)(32,40)(34,39)(35,38);(1,4,5,2)(3,7,8,6)(9,13,14,11)(10,15,12,16)(25,29)(26,32)(27,34)(28,35)(30,38)(31,39)(33,40)(36,37)Polycyclic presentation:
2^9|a^2=fghi,b^2=fgh,c^2=fgi,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5626250);