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 | |
4425 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,20)(12,16)(13,17)(14,18)(15,19)(21,22)(23,24)(25,26,30,31)(27,29,34,32)(28,36,38,33)(35,39,40,37);(1,3,5,8)(2,6,7,4)(11,12)(16,20)(21,23)(22,24)(26,31)(27,34)(33,36)(35,40);(1,3)(2,4)(5,8)(6,7)(9,11,15,12)(10,16,19,20)(13,21,18,23)(14,22,17,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,4)(2,3)(5,6)(7,8)(9,12,15,11)(10,16,19,20)(13,23,18,21)(14,22,17,24)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(9,13,15,18)(10,17,19,14)(11,21,12,23)(16,24,20,22)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(9,14,15,17)(10,18,19,13)(11,22,12,24)(16,23,20,21)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=gi,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8345010);