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 | #38035 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4271 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)(10,15)(11,13)(17,20)(19,21)(25,26)(27,30)(28,36)(29,34)(31,32)(33,39)(35,37)(38,40);(1,2,5,6)(3,7,4,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,22)(21,24)(26,32)(27,34)(33,36)(35,40);(1,3,5,4)(2,7,6,8)(9,11,14,13)(10,16,15,18)(12,19,22,21)(17,23,20,24)(25,27)(26,29)(28,35)(30,32)(31,34)(33,37)(36,38)(39,40);(9,10)(11,16)(12,20)(13,18)(14,15)(17,22)(19,24)(21,23)(25,28,31,39)(26,33,32,36)(27,35,34,40)(29,37,30,38);(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24)(25,29,31,30)(26,34,32,27)(28,37,39,38)(33,40,36,35);(1,4,5,3)(2,8,6,7)(9,13,14,11)(10,18,15,16)(12,21,22,19)(17,24,20,23)(25,30,31,29)(26,27,32,34)(28,38,39,37)(33,35,36,40)Polycyclic presentation:
2^9|b^2=gh,c^2=gi,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7554244);