Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10)(11,12)(13,15)(14,16)(17,18,21,20)(19,22,23,24)(25,26)(27,33)(28,34)(29,37)(30,31)(32,40)(35,36)(38,39);(1,3,5,8)(2,4,6,7)(9,11,14,15)(10,13,16,12)(19,23)(22,24)(26,31)(27,34)(32,37)(35,38);(3,8)(4,7)(9,12,14,13)(10,11,16,15)(17,19)(18,22)(20,24)(21,23)(25,27)(26,28)(29,38)(30,34)(31,33)(32,39)(35,40)(36,37);(1,2)(3,7)(4,8)(5,6)(9,13,14,12)(10,15,16,11)(17,20,21,18)(19,24,23,22)(25,28)(26,27)(29,39)(30,33)(31,34)(32,38)(35,37)(36,40);(1,4,5,7)(2,3,6,8)(9,10)(11,13)(12,15)(14,16)(25,29)(26,32)(27,35)(28,36)(30,40)(31,37)(33,39)(34,38)Polycyclic presentation:
2^9|a^2=fghi,b^2=fg,c^2=fi,d^2=hi,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3742288);