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 | #37788 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4392 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)(3,4)(5,6)(7,8)(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,26,30,32)(27,35,36,31)(28,37,39,33)(29,34,38,40);(2,6)(3,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,27,30,36)(26,31,32,35)(28,38,39,29)(33,40,37,34);(1,3,5,7)(2,4,6,8)(11,17)(14,21)(18,23)(22,24)(26,32)(31,35)(33,37)(34,40);(1,4)(2,7)(3,6)(5,8)(9,10)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,28,30,39)(26,33,32,37)(27,29,36,38)(31,34,35,40);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,26)(27,31)(28,33)(29,34)(30,32)(35,36)(37,39)(38,40);(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,23)(21,24)(25,29)(26,34)(27,28)(30,38)(31,33)(32,40)(35,37)(36,39)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=i,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8435318);