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 | |
4443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,13,16)(11,14,18,21)(12,15,19,22)(17,20,23,24)(25,26,31,35)(27,36,30,32)(28,34,29,33)(37,39,38,40);(1,2,5,4)(3,7,8,6)(9,11,13,18)(10,14,16,21)(12,17,19,23)(15,20,22,24)(25,27,31,30)(26,32,35,36)(28,37,29,38)(33,39,34,40);(2,4)(6,7)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,28,31,29)(26,33,35,34)(27,37,30,38)(32,39,36,40);(9,11)(10,14)(12,17)(13,18)(15,20)(16,21)(19,23)(22,24)(25,26)(27,36)(28,33)(29,34)(30,32)(31,35)(37,40)(38,39);(1,3,5,8)(2,6,4,7)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,29,31,28)(26,34,35,33)(27,37,30,38)(32,39,36,40);(1,4,5,2)(3,7,8,6)(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,20,21,24)(25,30,31,27)(26,32,35,36)(28,37,29,38)(33,40,34,39)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=g,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8162051);