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 | #45002 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4541 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,15,12)(11,18,16,13)(14,17,23,20)(19,22,24,21)(26,32)(27,29)(33,39)(35,36);(1,2,5,6)(3,7,4,8)(9,11)(10,13)(12,18)(14,19)(15,16)(17,21)(20,22)(23,24)(25,26)(27,30)(28,33)(29,34)(31,32)(35,37)(36,40)(38,39);(1,3,5,4)(2,7,6,8)(10,12)(11,16)(17,20)(19,24)(25,27,31,29)(26,30,32,34)(28,36,38,35)(33,40,39,37);(9,12,15,10)(11,18,16,13)(14,20,23,17)(19,22,24,21)(25,28,31,38)(26,33,32,39)(27,35,29,36)(30,37,34,40);(9,13,15,18)(10,16,12,11)(14,22,23,21)(17,19,20,24)(25,29,31,27)(26,34,32,30)(28,36,38,35)(33,40,39,37);(1,4,5,3)(2,8,6,7)(9,14)(10,17)(11,19)(12,20)(13,21)(15,23)(16,24)(18,22)(25,30,31,34)(26,27,32,29)(28,37,38,40)(33,35,39,36)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=gi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7790208);