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 | |
4345 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)(3,4)(5,6)(7,8)(11,17)(15,18)(19,21)(23,24)(25,26,31,30)(27,32,35,38)(28,37,39,33)(29,36,40,34);(2,6)(3,7)(9,10)(11,18)(12,20)(13,14)(15,17)(16,22)(19,23)(21,24)(25,27)(26,32)(28,36)(29,33)(30,38)(31,35)(34,39)(37,40);(1,3,5,7)(2,4,6,8)(10,14)(15,18)(16,20)(23,24)(26,30)(27,35)(33,37)(34,36);(1,4)(2,7)(3,6)(5,8)(9,11,13,17)(10,15,14,18)(12,21,22,19)(16,24,20,23)(25,28)(26,33)(27,36)(29,32)(30,37)(31,39)(34,35)(38,40);(9,12,13,22)(10,16,14,20)(11,19,17,21)(15,23,18,24)(25,29,31,40)(26,34,30,36)(27,37,35,33)(28,38,39,32);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,30,31,26)(27,38,35,32)(28,37,39,33)(29,36,40,34)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8324038);