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 |
| prime radical | #2408 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,26,30,35)(27,31,37,32)(28,38,39,33)(29,34,40,36);(1,2,4,5)(3,6,7,8)(25,27)(26,31)(28,36)(29,38)(30,37)(32,35)(33,40)(34,39);(10,15)(11,16)(17,22)(18,23)(25,27,30,37)(26,32,35,31)(28,34,39,36)(29,33,40,38);(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,27)(26,32)(28,36)(29,33)(30,37)(31,35)(34,39)(38,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,28)(26,33)(27,36)(29,32)(30,39)(31,40)(34,37)(35,38);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,29)(26,34)(27,33)(28,31)(30,40)(32,39)(35,36)(37,38)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=i,d^2=hi,e^2=g,f^2=hi,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, 7894514);