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 | #54609 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,26)(27,36)(28,32)(29,39)(30,35)(31,37)(33,38)(34,40);(2,5)(6,7)(10,14)(15,17)(16,21)(22,23)(25,27,30,37)(26,31,35,36)(28,34,38,39)(29,33,40,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28,30,38)(26,32,35,33)(27,34,37,39)(29,36,40,31);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28)(26,33)(27,34)(29,36)(30,38)(31,40)(32,35)(37,39);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,26)(27,31)(28,32)(29,34)(30,35)(33,38)(36,37)(39,40);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29)(26,34)(27,33)(28,36)(30,40)(31,38)(32,37)(35,39)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066247);