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 | #34727 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,15,17)(11,16,12,14)(13,21,22,18)(19,24,20,23)(27,33)(31,35)(34,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,17,14)(13,19,22,20)(18,23,21,24)(26,29)(31,35)(32,36)(38,40);(10,17)(11,12)(13,22)(23,24)(25,26,30,29)(27,31,33,35)(28,36,37,32)(34,40,39,38);(1,3,4,7)(2,6,5,8)(9,12,15,11)(10,16,17,14)(13,20,22,19)(18,23,21,24)(25,27,30,33)(26,31,29,35)(28,34,37,39)(32,38,36,40);(9,13)(10,18)(11,19)(12,20)(14,24)(15,22)(16,23)(17,21)(25,28,30,37)(26,32,29,36)(27,34,33,39)(31,38,35,40);(9,14,15,16)(10,11,17,12)(13,23,22,24)(18,20,21,19)(25,29,30,26)(27,35,33,31)(28,32,37,36)(34,38,39,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=hi,d^2=gi,e^2=hi,f^2=i,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, 7896826);