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 | |
4542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(11,17)(12,20)(15,18)(16,21)(25,26,30,31)(27,29,33,40)(28,35,37,32)(34,38,36,39);(1,3,5,8)(2,6,7,4)(9,10)(11,18)(12,21)(13,14)(15,17)(16,20)(19,23)(22,24)(25,27)(26,29)(28,36)(30,33)(31,40)(32,38)(34,37)(35,39);(1,3)(2,4)(5,8)(6,7)(10,14)(11,17)(16,21)(19,22)(26,31)(27,33)(28,37)(38,39);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,22)(13,17)(14,18)(16,24)(19,20)(21,23)(25,28,30,37)(26,32,31,35)(27,34,33,36)(29,39,40,38);(9,12)(10,16)(11,19)(13,20)(14,21)(15,23)(17,22)(18,24)(25,29)(26,27)(28,38)(30,40)(31,33)(32,36)(34,35)(37,39);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,26)(27,29)(28,32)(30,31)(33,40)(34,39)(35,37)(36,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8840310);