Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667748 |
| 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 | |
2690 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)(12,18)(13,17)(14,19)(15,20)(21,30)(22,29)(23,31)(24,32)(25,34)(26,33)(27,36)(28,35)(37,40)(38,39);(1,3,4,8)(2,5,6,7)(9,11,15,24)(10,16,20,32)(12,25,27,37)(13,23,14,21)(17,31,19,30)(18,34,36,40)(22,28,38,26)(29,35,39,33);(10,17)(11,21)(13,14)(18,33)(19,20)(22,25)(23,24)(26,28)(30,31)(34,40)(35,36)(37,38);(2,6)(5,7);(3,8)(5,7)(9,12,15,27)(10,18,20,36)(11,22,24,38)(13,26,14,28)(16,29,32,39)(17,33,19,35)(21,25,23,37)(30,34,31,40)Polycyclic presentation:
2^9|b^2=fi,e^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,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6575599);