Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682658 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1791 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,13)(14,18)(15,27)(16,19)(20,24)(21,25)(22,32)(23,33)(26,28)(29,34)(30,35)(31,36)(37,39)(38,40);(2,5)(3,7)(9,11,16,24)(10,17,19,20)(12,25,27,33)(13,21,15,23)(14,29,31,38)(18,34,36,40)(22,28,39,30)(26,37,35,32);(9,12,13,10,16,27,15,19)(11,20,21,25,24,17,23,33)(14,26,28,18,31,35,30,36)(22,37,29,34,39,32,38,40);(2,5)(6,8)(9,13,16,15)(10,12,19,27)(11,21,24,23)(14,28,31,30)(17,25,20,33)(18,26,36,35)(22,29,39,38)(32,34,37,40);(1,3,4,7)(2,6,5,8)(9,14)(10,18)(11,22)(12,26)(13,28)(15,30)(16,31)(17,32)(19,36)(20,37)(21,29)(23,38)(24,39)(25,34)(27,35)(33,40)Polycyclic presentation:
2^9|b^2=i,c^2=gi,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592717);