Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1640 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)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,20)(21,23)(22,24)(25,26,29,32)(27,30,35,33)(28,36,37,31)(34,39,40,38);(2,6)(3,7)(9,11)(10,12)(13,16)(14,15)(17,19)(18,22)(20,23)(21,24)(25,26,29,32)(27,33,35,30)(28,31,37,36)(34,39,40,38);(2,6)(4,8)(10,14)(11,16)(17,19,21,24)(18,22,23,20)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(1,3,5,7)(2,4,6,8)(9,12,13,15)(10,11,14,16)(17,20)(18,19)(21,22)(23,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,4)(2,7)(3,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,19)(18,20)(21,24)(22,23)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fghi,d^2=ghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4611795);