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 | #59303 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10,15,19)(11,16,21,24)(12,13,17,22)(14,23,20,18)(25,26)(27,33)(28,29)(30,38)(31,32)(34,39)(35,36)(37,40);(1,2,5,4)(3,7,8,6)(9,11,15,21)(10,16,19,24)(12,20,17,14)(13,18,22,23)(26,32)(28,33)(34,38)(36,40);(1,3,5,8)(2,6,4,7)(9,12,15,17)(10,13,19,22)(11,14,21,20)(16,23,24,18)(25,27,31,29)(26,28,32,33)(30,35,39,37)(34,36,38,40);(1,4,5,2)(3,7,8,6)(9,12)(10,13)(11,20)(14,21)(15,17)(16,18)(19,22)(23,24)(25,28)(26,27)(29,32)(30,36)(31,33)(34,35)(37,38)(39,40);(9,13)(10,17)(11,18)(12,19)(14,16)(15,22)(20,24)(21,23)(25,29,31,27)(26,33,32,28)(30,37,39,35)(34,40,38,36);(9,14,15,20)(10,18,19,23)(11,12,21,17)(13,16,22,24)(25,30,31,39)(26,34,32,38)(27,35,29,37)(28,36,33,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=hi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8081576);