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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4269 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)(11,19)(12,16)(13,17)(14,18)(15,20)(21,24)(22,23)(25,26,31,35)(27,36,39,32)(28,38,37,33)(29,34,40,30);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,17)(18,19)(23,24);(11,20)(13,21)(15,19)(17,24)(25,27,31,39)(26,32,35,36)(28,29,37,40)(30,33,34,38);(1,3,5,4)(2,7,6,8)(9,12,14,22)(10,16,18,23)(11,21,20,13)(15,17,19,24)(25,28)(26,33)(27,29)(30,36)(31,37)(32,34)(35,38)(39,40);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,23)(13,17)(14,18)(16,22)(19,20)(21,24)(25,29)(26,34)(27,37)(28,39)(30,35)(31,40)(32,38)(33,36);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,17,18,24)(11,22,20,12)(15,16,19,23)(25,30,31,34)(26,29,35,40)(27,38,39,33)(28,36,37,32)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=h,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7558925);