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 | |
4330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(10,11)(13,21)(15,17)(20,24)(25,26,31,32)(27,28,38,33)(29,34,30,35)(36,39,37,40);(1,2,5,4)(3,7,8,6)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(26,32)(28,33)(34,35)(39,40);(1,3,5,8)(2,6,4,7)(12,19)(15,17)(20,24)(22,23)(25,27)(26,33)(28,32)(29,36)(30,37)(31,38)(34,40)(35,39);(1,4,5,2)(3,7,8,6)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28,31,33)(26,27,32,38)(29,39,30,40)(34,36,35,37);(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,29,31,30)(26,34,32,35)(27,36,38,37)(28,39,33,40);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,29)(26,35,32,34)(27,37,38,36)(28,40,33,39)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=i,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8002409);