Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1725 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,6,7,8)(9,10,13,14)(11,12,16,15)(17,18,21,20)(19,23,24,22)(26,30)(27,33)(28,36)(38,40);(2,5)(3,7)(10,14)(12,15)(18,20)(22,23);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,14)(13,16)(17,19,21,24)(18,22,20,23)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(17,18,21,20)(19,22,24,23)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,11,14,16)(17,20,21,18)(19,23,24,22)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=ghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6018140);