Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1691 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,13,14)(11,12,16,15)(18,20)(22,23)(25,26,29,30)(27,31,35,33)(28,36,37,32)(34,40,38,39);(4,8)(6,7)(9,11,13,16)(10,12,14,15)(18,20)(22,23)(26,30)(28,37)(31,33)(34,38);(2,3)(6,7)(10,14)(12,15)(17,18,21,20)(19,23,24,22)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,39)(38,40);(1,3,5,2)(4,7,8,6)(9,12,13,15)(10,11,14,16)(17,19,21,24)(18,22,20,23)(25,27)(26,31)(28,38)(29,35)(30,33)(32,40)(34,37)(36,39);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,20,21,18)(19,23,24,22)(25,28)(26,32)(27,34)(29,37)(30,36)(31,39)(33,40)(35,38)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=fh,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4086934);