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 | |
4341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(11,17)(15,18)(19,23)(22,24)(25,26)(27,31)(28,38)(29,40)(30,32)(33,39)(34,36)(35,37);(2,3)(6,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27)(26,31)(28,29)(30,35)(32,37)(33,34)(36,39)(38,40);(1,3,5,2)(4,7,8,6)(10,14)(15,18)(16,20)(22,24)(26,32)(27,35)(28,39)(34,40);(1,4)(2,6)(3,7)(5,8)(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,28)(26,33)(27,36)(29,35)(30,39)(31,40)(32,38)(34,37);(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(25,26)(27,37)(28,33)(29,40)(30,32)(31,35)(34,36)(38,39);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29,30,36)(26,34,32,40)(27,28,35,39)(31,33,37,38)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=ghi,f^2=h,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, 9945695);