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 | #59812 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4465 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,17)(13,21)(14,16)(18,22)(19,20)(23,24)(26,31)(27,28)(34,37)(35,36);(2,3)(6,7)(9,11,14,17)(10,15,16,12)(13,19,22,24)(18,23,21,20)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(4,8)(6,7)(10,16)(11,17)(18,21)(19,24)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,40)(36,39);(1,3,5,2)(4,7,8,6)(9,12,14,15)(10,17,16,11)(13,20,22,23)(18,24,21,19)(25,27,30,28)(26,32,31,33)(29,35,38,36)(34,39,37,40);(9,13,14,22)(10,18,16,21)(11,19,17,24)(12,20,15,23)(25,28,30,27)(26,33,31,32)(29,36,38,35)(34,40,37,39);(1,4,5,8)(2,6,3,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,24)(21,22)(25,29,30,38)(26,34,31,37)(27,35,28,36)(32,39,33,40)Polycyclic presentation:
2^9|b^2=i,d^2=gi,e^2=h,f^2=g,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, 8095384);