Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,8)(4,7)(5,6)(25,26)(27,31)(28,32)(29,37)(30,34)(33,38)(35,36)(39,40);(1,3,6,4)(2,7,5,8)(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23);(1,2,6,5)(3,7,4,8)(10,12)(15,17)(16,20)(22,23)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,39)(38,40);(1,2,6,5)(3,8,4,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27)(26,31)(28,34)(29,35)(30,32)(33,39)(36,37)(38,40);(1,4,6,3)(2,7,5,8)(9,12,14,10)(11,15,19,17)(13,20,21,16)(18,22,24,23)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,40,37,35);(1,5,6,2)(3,7,4,8)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,29,30,38)(26,33,34,37)(27,35,32,40)(28,36,31,39)Polycyclic presentation:
2^9|b^2=gi,c^2=g,d^2=gi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10314989);