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 | #41957 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4227 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)(11,13)(12,17)(14,16)(15,18)(19,22)(20,21)(23,24)(27,29)(31,33)(35,37)(39,40);(1,2,5,4)(3,7,8,6)(9,11,14,18)(10,15,16,13)(12,19,20,24)(17,23,21,22)(25,26,30,34)(27,31,29,33)(28,36,38,32)(35,40,37,39);(1,3,5,8)(2,6,4,7)(10,16)(12,20)(13,15)(19,24)(25,27,30,29)(26,31,34,33)(28,35,38,37)(32,39,36,40);(1,4,5,2)(3,7,8,6)(9,12,14,20)(10,17,16,21)(11,19,18,24)(13,22,15,23);(9,10)(11,15)(12,21)(13,18)(14,16)(17,20)(19,22)(23,24)(25,28,30,38)(26,32,34,36)(27,35,29,37)(31,39,33,40);(9,13,14,15)(10,18,16,11)(12,22,20,23)(17,24,21,19)(25,29,30,27)(26,33,34,31)(28,37,38,35)(32,40,36,39)Polycyclic presentation:
2^9|b^2=hi,c^2=gh,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8961603);