Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
2589 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)(11,20)(12,26)(13,17)(14,19)(15,23)(16,29)(18,30)(21,38)(22,24)(25,36)(27,32)(28,35)(31,34)(33,39)(37,40);(1,3,4,8)(2,5,6,7)(9,11,13,23)(10,15,17,20)(12,27,28,40)(14,24,30,25)(16,33,34,38)(18,22,19,36)(21,31,39,29)(26,37,35,32);(9,12,13,28)(10,16,17,34)(11,21,23,39)(14,31,30,29)(15,32,20,37)(18,35,19,26)(22,38,36,33)(24,27,25,40);(2,6)(5,7)(10,17)(15,20)(16,34)(18,19)(22,36)(26,35)(32,37)(33,38);(3,8)(5,7)(9,10)(11,22)(12,16)(13,17)(14,18)(15,25)(19,30)(20,24)(21,38)(23,36)(26,29)(27,37)(28,34)(31,35)(32,40)(33,39)Polycyclic presentation:
2^9|b^2=f,c^2=i,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,g^c=gi,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, 6575812);