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 | #50646 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,16)(11,17,18,13)(12,15,22,24)(19,23,20,21)(25,26,30,31)(27,28,33,32)(29,37,39,34)(35,40,38,36);(1,3,5,8)(2,6,7,4)(9,10)(11,13)(12,15)(14,16)(17,18)(19,21)(20,23)(22,24)(26,31)(28,32)(34,37)(36,40);(1,3)(2,4)(5,8)(6,7)(11,18)(13,17)(19,20)(21,23)(25,27,30,33)(26,32,31,28)(29,38,39,35)(34,36,37,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,13)(12,20)(14,18)(15,23)(16,17)(19,22)(21,24)(25,27)(26,32)(28,31)(29,35)(30,33)(34,40)(36,37)(38,39);(9,12,14,22)(10,15,16,24)(11,19,18,20)(13,21,17,23)(25,28,30,32)(26,33,31,27)(29,36,39,40)(34,38,37,35);(9,13)(10,11)(12,21)(14,17)(15,19)(16,18)(20,24)(22,23)(25,29)(26,34)(27,35)(28,36)(30,39)(31,37)(32,40)(33,38)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8825022);