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 | |
4514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,12)(13,21)(14,15)(16,22)(17,19)(18,24)(20,23)(25,26,30,31)(27,28,35,32)(29,33,37,38)(34,36,40,39);(2,5)(3,7)(11,17)(12,19)(18,23)(20,24)(26,31)(28,32)(33,38)(36,39);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(16,21)(18,23)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,18)(14,17)(15,19)(16,20)(21,24)(22,23)(25,28,30,32)(26,27,31,35)(29,36,37,39)(33,34,38,40);(9,12)(10,11)(13,20)(14,19)(15,17)(16,18)(21,23)(22,24);(9,13)(10,16)(11,18)(12,20)(14,22)(15,21)(17,23)(19,24)(25,29,30,37)(26,33,31,38)(27,34,35,40)(28,36,32,39)Polycyclic presentation:
2^9|a^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000597);