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 | |
2154 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,26)(13,18)(14,19)(15,20)(17,29)(21,37)(22,23)(24,32)(25,33)(27,31)(28,35)(30,36)(34,38)(39,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,25)(10,16,20,33)(12,27,30,39)(13,24,14,22)(17,34,35,37)(18,32,19,23)(21,29,38,28)(26,31,36,40);(9,12,14,29,15,30,13,28)(10,17,18,26,20,35,19,36)(11,21,22,27,25,38,24,39)(16,31,32,37,33,40,23,34);(2,6)(5,7)(9,13,15,14)(10,18,20,19)(11,22,25,24)(12,28,30,29)(16,23,33,32)(17,26,35,36)(21,27,38,39)(31,34,40,37);(3,8)(5,7)(9,10)(11,23)(12,17)(13,19)(14,18)(15,20)(16,22)(21,34)(24,33)(25,32)(26,29)(27,31)(28,36)(30,35)(37,38)(39,40)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589859);