Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1758 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)(3,6)(4,8)(5,7)(9,10,13,14)(11,12,15,16)(19,23)(21,24)(25,26)(27,33)(28,30)(29,36)(31,34)(32,37)(35,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,10,13,14)(11,12,15,16)(17,18,20,22)(19,21,23,24)(26,28)(27,34)(29,37)(39,40);(2,7)(3,8)(10,14)(11,15)(17,18)(19,21)(20,22)(23,24)(25,27)(26,31)(28,33)(29,38)(30,34)(32,40)(35,37)(36,39);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,15)(14,16)(25,28,30,26)(27,33,34,31)(29,36,37,32)(35,40,38,39);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,19)(18,21)(20,23)(22,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,39)(33,40)(34,38)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 4971713);