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 | #52962 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,17)(12,18)(13,21)(14,15)(16,22)(19,20)(23,24)(29,37)(32,36)(34,40)(38,39);(1,3,5,8)(2,6,7,4)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(1,3)(2,4)(5,8)(6,7)(10,15)(11,18)(16,21)(19,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23);(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,28,30,26)(27,31,35,33)(29,36,37,32)(34,38,40,39);(9,13,14,22)(10,16,15,21)(11,19,18,24)(12,20,17,23)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002596);