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 | |
4536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,5,7,4)(9,10)(11,14)(12,19)(13,15)(16,20)(17,23)(18,21)(22,24)(25,26,30,28)(27,32,35,31)(29,33,38,36)(34,40,37,39);(1,3,6,7)(2,5,8,4)(9,11,13,18)(10,14,15,21)(12,17,20,24)(16,22,19,23)(25,27)(26,31)(28,32)(29,34)(30,35)(33,39)(36,40)(37,38);(1,4)(2,7)(3,8)(5,6)(10,15)(14,21)(16,19)(22,23)(25,26,30,28)(27,31,35,32)(29,33,38,36)(34,39,37,40);(1,2)(3,4)(5,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,20)(23,24)(25,28,30,26)(27,31,35,32)(29,36,38,33)(34,39,37,40);(1,5,6,4)(2,7,8,3)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,27)(26,32)(28,31)(29,37)(30,35)(33,39)(34,38)(36,40);(1,2)(3,5)(4,7)(6,8)(9,12)(10,16)(11,17)(13,20)(14,22)(15,19)(18,24)(21,23)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=g,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9962642);