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 | |
4635 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)(3,7)(4,6)(5,8)(9,10)(11,12)(13,21)(14,15)(16,17)(18,22)(19,23)(20,24)(26,27)(29,37)(31,33)(36,40);(1,2,4,6)(3,5,8,7)(10,15)(11,17)(18,21)(19,24)(25,26,30,27)(28,31,35,33)(29,32,37,34)(36,38,40,39);(1,2)(3,5)(4,6)(7,8)(10,15)(12,16)(18,21)(20,23)(28,35)(31,33)(36,40)(38,39);(1,3,4,8)(2,5,6,7)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,23,21,20)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,39,40,38);(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39);(9,13,14,22)(10,18,15,21)(11,19,17,24)(12,20,16,23)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|b^2=gh,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8027092);