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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,14,15)(11,13,18,16)(12,20,23,17)(19,24,21,22)(26,31)(27,32)(33,38)(34,39);(2,3)(6,7)(10,15)(11,18)(17,20)(19,21)(25,26,30,31)(27,28,32,36)(29,33,37,38)(34,35,39,40);(1,3,5,2)(4,7,8,6)(10,15)(13,16)(17,20)(22,24)(29,37)(33,38)(34,39)(35,40);(1,4,5,8)(2,6,3,7)(9,11,14,18)(10,16,15,13)(12,21,23,19)(17,22,20,24)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,39)(38,40);(9,12,14,23)(10,17,15,20)(11,19,18,21)(13,22,16,24)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(9,13,14,16)(10,18,15,11)(12,22,23,24)(17,21,20,19)(25,29,30,37)(26,33,31,38)(27,34,32,39)(28,35,36,40)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=i,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8247031);