Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C23×C43 |
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 | |
1872 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)(4,6,8,7)(9,10,14,15)(11,13,12,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,39)(38,40);(4,8)(6,7)(9,11,14,12)(10,13,15,16)(17,19,21,24)(18,22,20,23)(26,31)(27,32)(29,37)(35,40);(2,3)(6,7)(10,15)(11,12)(18,20)(22,23)(25,27,30,32)(26,28,31,36)(29,38,37,34)(33,40,39,35);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,13,15,16)(17,20,21,18)(19,23,24,22)(25,28)(26,32)(27,31)(29,35)(30,36)(33,38)(34,39)(37,40);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,15)(14,16)(17,19)(18,22)(20,23)(21,24)(25,29,30,37)(26,33,31,39)(27,34,32,38)(28,35,36,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=fi,c^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5393694);