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 | |
4979 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(11,17)(15,18)(19,23)(22,24)(25,26)(27,33)(28,35)(29,37)(30,31)(32,39)(34,36)(38,40);(1,3,5,8)(2,6,7,4)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(25,27,30,35)(26,28,31,33)(29,38,39,34)(32,40,37,36);(1,3)(2,4)(5,8)(6,7)(10,14)(15,18)(16,20)(22,24)(26,31)(28,33)(32,37)(36,40);(1,4)(2,3)(5,6)(7,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28,30,33)(26,27,31,35)(29,36,39,40)(32,34,37,38);(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29)(26,32)(27,34)(28,36)(30,39)(31,37)(33,40)(35,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8731928);