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 | #37189 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4254 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,32,36)(27,33,38,28)(29,40,37,34)(30,35,39,31);(1,2)(3,7)(4,8)(5,6)(25,27)(26,33)(28,36)(29,35)(30,34)(31,37)(32,38)(39,40);(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,28)(26,27)(29,39)(30,37)(31,40)(32,33)(34,35)(36,38);(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,29,32,37)(26,34,36,40)(27,35,38,31)(28,39,33,30);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,30,32,39)(26,35,36,31)(27,34,38,40)(28,37,33,29);(1,4,5,3)(2,8,6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,31,32,35)(26,30,36,39)(27,37,38,29)(28,40,33,34)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=i,e^2=gi,f^2=ghi,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, 7823460);