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 | #38509 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,15)(12,22)(13,18)(14,16)(17,19)(20,21)(23,24)(25,26,32,36)(27,37,38,33)(28,34,29,39)(30,31,35,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,20)(13,23)(14,19)(16,17)(18,24)(21,22)(25,27,32,38)(26,33,36,37)(28,35,29,30)(31,39,40,34);(1,3,5,4)(2,6,8,7)(10,16)(11,19)(12,18)(21,23)(25,28,32,29)(26,34,36,39)(27,30,38,35)(31,33,40,37);(1,4,5,3)(2,7,8,6)(9,11)(10,17)(12,20)(13,21)(14,19)(15,16)(18,24)(22,23)(25,29,32,28)(26,34,36,39)(27,30,38,35)(31,37,40,33);(9,12)(10,13)(11,20)(14,18)(15,23)(16,22)(17,21)(19,24)(25,30)(26,31)(27,28)(29,38)(32,35)(33,39)(34,37)(36,40);(9,13)(10,18)(11,21)(12,16)(14,22)(15,20)(17,24)(19,23)(25,31)(26,35)(27,34)(28,33)(29,37)(30,36)(32,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7876638);