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 | #36426 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,17)(12,21)(13,14)(15,16)(18,24)(19,20)(22,23)(25,26,30,33)(27,28,35,31)(29,37,38,32)(34,39,40,36);(2,6)(3,7)(10,16)(13,24)(17,21)(19,22)(25,27,30,35)(26,31,33,28)(29,34,38,40)(32,39,37,36);(1,3)(2,4)(5,7)(6,8)(9,11,15,12)(10,17,16,21)(13,19,24,22)(14,20,18,23)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,37)(38,40);(1,4)(2,3)(5,8)(6,7)(9,12,15,11)(10,17,16,21)(13,22,24,19)(14,20,18,23)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,37,40);(9,13)(10,18)(11,19)(12,22)(14,16)(15,24)(17,23)(20,21)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,37)(35,40);(9,14,15,18)(10,13,16,24)(11,20,12,23)(17,19,21,22)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gi,d^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8412207);