Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1734 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)(3,7)(4,5)(6,8)(9,10,13,12)(11,14,16,15)(17,18,20,22)(19,21,23,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(2,4)(3,8)(9,10)(11,15)(12,13)(14,16)(19,23)(21,24)(25,26,29,32)(27,33,35,30)(28,31,37,36)(34,39,40,38);(2,4)(6,7)(10,12)(14,15)(17,19,20,23)(18,21,22,24)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,15,16,14)(17,18)(19,21)(20,22)(23,24)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=fhi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5156818);