Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667760 |
| 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 | |
2024 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)(11,21)(12,27)(13,20)(14,19)(15,18)(16,25)(17,29)(22,38)(23,34)(24,26)(28,33)(30,32)(31,36)(35,40)(37,39);(2,5)(3,7)(9,11,14,25)(10,16,19,21)(12,28,31,39)(13,23,15,26)(17,35,30,38)(18,34,20,24)(22,32,40,29)(27,37,36,33);(1,3,4,7)(2,6,5,8)(9,12,14,31)(10,17,19,30)(11,22,25,40)(13,29,15,32)(16,33,21,37)(18,27,20,36)(23,39,26,28)(24,38,34,35);(2,5)(6,8)(9,13,14,15)(10,18,19,20)(11,23,25,26)(12,29,31,32)(16,34,21,24)(17,27,30,36)(22,39,40,28)(33,35,37,38);(9,10)(11,24)(12,30)(13,18)(14,19)(15,20)(16,23)(17,31)(21,26)(22,35)(25,34)(27,32)(28,33)(29,36)(37,39)(38,40)Polycyclic presentation:
2^9|b^2=i,c^2=f,d^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,e^c=ei,g^c=gi,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, 6576262);