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 | |
4353 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(10,14)(12,20)(15,17)(18,21)(25,26)(27,31)(28,37)(29,33)(30,34)(32,39)(35,36)(38,40);(1,2,5,4)(3,7,8,6)(10,14)(15,17)(16,22)(23,24)(25,27,30,36)(26,31,34,35)(28,38,39,29)(32,33,37,40);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,36)(37,39)(38,40);(1,4,5,2)(3,7,8,6)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,28)(26,32)(27,29)(30,39)(31,40)(33,35)(34,37)(36,38);(9,11,13,19)(10,15,14,17)(12,21,20,18)(16,24,22,23)(25,26)(27,31)(28,32)(29,40)(30,34)(33,38)(35,36)(37,39);(9,12)(10,16)(11,18)(13,20)(14,22)(15,23)(17,24)(19,21)(25,29)(26,33)(27,28)(30,38)(31,37)(32,35)(34,40)(36,39)Polycyclic presentation:
2^9|b^2=hi,c^2=g,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10367922);