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 | |
4264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,15)(11,12,17,20)(13,16,21,22)(18,19,23,24)(25,26)(27,32)(28,37)(29,36)(30,35)(31,33)(34,38)(39,40);(1,2,5,6)(3,7,4,8)(25,27)(26,32)(28,30)(29,33)(31,36)(34,40)(35,37)(38,39);(1,3,5,4)(2,7,6,8)(10,15)(11,17)(16,22)(18,23)(26,33)(27,36)(28,38)(35,40);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,28)(26,34)(27,30)(29,35)(31,38)(32,40)(33,37)(36,39);(1,4,5,3)(2,8,6,7)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,29,31,32)(26,27,33,36)(28,35,38,40)(30,37,39,34);(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,30)(26,35)(27,28)(29,34)(31,39)(32,37)(33,40)(36,38)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=g,d^2=g,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7806430);