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 | #34976 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,13)(12,15)(14,22)(16,18)(17,24)(19,20)(21,23)(25,26,30,33)(27,31,35,28)(29,32,38,40)(34,39,37,36);(2,6)(3,7)(9,11,16,15)(10,13,18,12)(14,19,24,23)(17,21,22,20)(25,27,30,35)(26,31,33,28)(29,34,38,37)(32,39,40,36);(1,3)(2,4)(5,7)(6,8)(9,12,16,13)(10,11,18,15)(14,20,24,21)(17,19,22,23)(25,27)(26,31)(28,33)(29,37)(30,35)(32,36)(34,38)(39,40);(1,4)(2,3)(5,8)(6,7)(9,13,16,12)(10,11,18,15)(14,21,24,20)(17,19,22,23)(25,28,30,31)(26,27,33,35)(29,36,38,39)(32,34,40,37);(9,14,16,24)(10,17,18,22)(11,19,15,23)(12,20,13,21);(9,15,16,11)(10,12,18,13)(14,23,24,19)(17,20,22,21)(25,29,30,38)(26,32,33,40)(27,34,35,37)(28,36,31,39)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=h,d^2=gi,e^2=h,f^2=gi,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, 8104214);