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 | |
4567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,17)(11,19,13,16)(12,14,20,18)(21,22,23,24)(25,26,31,30)(27,35,36,32)(28,39,38,33)(29,34,37,40);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,16,17,19)(12,21,20,23)(14,24,18,22)(25,27)(26,32)(28,37)(29,38)(30,35)(31,36)(33,40)(34,39);(1,2,5,4)(3,6,8,7)(10,17)(11,13)(12,20)(22,24)(27,36)(29,37)(32,35)(34,40);(1,2,5,4)(3,6,8,7)(9,12,15,20)(10,18,17,14)(11,21,13,23)(16,22,19,24)(25,28,31,38)(26,33,30,39)(27,37,36,29)(32,40,35,34);(1,4,5,2)(3,7,8,6)(9,13,15,11)(10,19,17,16)(12,23,20,21)(14,22,18,24)(25,29,31,37)(26,34,30,40)(27,38,36,28)(32,39,35,33);(9,14,15,18)(10,20,17,12)(11,22,13,24)(16,21,19,23)(25,30,31,26)(27,32,36,35)(28,39,38,33)(29,34,37,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gi,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7893197);