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 | #44272 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4345 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,26)(27,35)(28,29)(30,31)(32,38)(33,34)(36,37)(39,40);(1,2,5,3)(4,7,8,6)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,22)(23,24)(26,31)(29,33)(32,35)(37,40);(2,3)(6,7)(9,10,13,16)(11,17,19,14)(12,15,21,20)(18,23,24,22)(25,27,30,38)(26,32,31,35)(28,39,34,36)(29,37,33,40);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(25,28,30,34)(26,33,31,29)(27,36,38,39)(32,37,35,40);(1,3,5,2)(4,7,8,6)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,26)(27,35)(28,33)(29,34)(30,31)(32,38)(36,40)(37,39);(1,4,5,8)(2,6,3,7)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,30,33)(26,34,31,28)(27,37,38,40)(32,36,35,39)Polycyclic presentation:
2^9|b^2=ghi,c^2=h,d^2=hi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8730982);