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 | #52299 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,18)(17,24)(19,21)(26,31)(28,32)(34,40)(36,38);(1,2)(3,6)(4,7)(5,8)(9,10,14,15)(11,16,18,12)(13,17,23,24)(19,22,21,20)(26,31)(27,33)(34,40)(35,37);(1,3)(2,4)(5,7)(6,8)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,37)(39,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,15)(13,21)(14,18)(17,20)(19,23)(22,24)(25,27,30,33)(26,32,31,28)(29,37,39,35)(34,36,40,38);(1,3)(2,6)(4,8)(5,7)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,21,24,19)(25,28,30,32)(26,33,31,27)(29,38,39,36)(34,35,40,37);(1,4,5,6)(2,7,8,3)(9,13,14,23)(10,17,15,24)(11,19,18,21)(12,20,16,22)(25,29,30,39)(26,34,31,40)(27,35,33,37)(28,36,32,38)Polycyclic presentation:
2^9|b^2=g,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^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8584595);