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 | #39004 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,20)(12,22)(13,14)(15,17)(16,21)(18,24)(19,23)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,2,5,4)(3,6,7,8)(9,11,15,21)(10,16,17,20)(12,23,24,13)(14,18,19,22)(26,27)(31,33)(32,34)(38,39);(2,4)(6,8)(10,17)(14,19)(16,20)(18,22)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(3,7)(6,8)(9,12,15,24)(10,18,17,22)(11,13,21,23)(14,20,19,16)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,19)(11,12)(14,17)(15,23)(16,22)(18,20)(21,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39);(1,4,5,2)(3,8,7,6)(9,14,15,19)(10,13,17,23)(11,18,21,22)(12,20,24,16)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gh,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8232720);