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 | |
4369 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)(3,7)(4,5)(6,8)(9,10,14,15)(11,16,21,23)(12,13,18,17)(19,24,22,20)(27,33)(31,34)(35,37)(39,40);(2,5)(6,7)(10,15)(12,18)(16,23)(19,22)(25,26)(27,34)(28,36)(29,30)(31,33)(32,38)(35,39)(37,40);(1,2)(3,6)(4,5)(7,8)(10,15)(13,17)(16,23)(20,24)(26,30)(31,34)(32,36)(39,40);(1,3)(2,6)(4,8)(5,7)(9,11,14,21)(10,16,15,23)(12,22,18,19)(13,20,17,24)(25,27,29,33)(26,31,30,34)(28,37,38,35)(32,40,36,39);(9,12,14,18)(10,17,15,13)(11,19,21,22)(16,20,23,24)(25,28,29,38)(26,32,30,36)(27,35,33,37)(31,39,34,40);(9,13,14,17)(10,18,15,12)(11,20,21,24)(16,19,23,22)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=h,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8323978);