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 | |
1935 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)(11,15)(12,25)(13,17)(14,19)(16,27)(18,30)(20,37)(21,32)(22,23)(24,33)(26,31)(28,36)(29,35)(34,39)(38,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,26)(13,23)(14,24)(16,34)(17,22)(18,32)(19,33)(20,28)(21,30)(25,31)(27,39)(29,38)(35,40)(36,37);(9,12,13,27,14,29,30,28)(10,16,18,25,19,36,17,35)(11,20,23,26,24,39,21,38)(15,31,32,37,33,40,22,34);(2,6)(5,7)(9,13,14,30)(10,17,19,18)(11,21,24,23)(12,27,29,28)(15,32,33,22)(16,35,36,25)(20,38,39,26)(31,37,40,34);(3,8)(5,7)(9,10)(11,22)(12,16)(13,18)(14,19)(15,23)(17,30)(20,34)(21,33)(24,32)(25,27)(26,31)(28,35)(29,36)(37,39)(38,40)Polycyclic presentation:
2^9|b^2=f,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, 6590476);