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 | #53101 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,4)(7,8)(9,10,15,12)(11,16,13,17)(14,18,22,20)(19,23,21,24)(25,26,30,34)(27,32,28,31)(29,33,38,37)(35,40,36,39);(1,2,6,5)(3,8,4,7)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,40)(37,38);(1,3,6,4)(2,7,5,8)(9,11,15,13)(10,17,12,16)(14,19,22,21)(18,24,20,23)(25,26)(27,31)(28,32)(29,37)(30,34)(33,38)(35,40)(36,39);(9,12,15,10)(11,16,13,17)(14,20,22,18)(19,23,21,24)(25,27,30,28)(26,31,34,32)(29,35,38,36)(33,39,37,40);(1,4,6,3)(2,7,5,8)(9,13,15,11)(10,16,12,17)(14,21,22,19)(18,23,20,24)(25,28,30,27)(26,32,34,31)(29,36,38,35)(33,40,37,39);(1,5,6,2)(3,7,4,8)(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gh,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8095278);