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 | |
4240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,16)(12,22)(13,17)(14,18)(15,19)(20,23)(21,24)(26,30)(28,35)(31,36)(34,39);(1,3,5,8)(2,6,7,4)(9,11)(10,16)(12,21)(13,18)(14,17)(15,20)(19,23)(22,24)(27,33)(31,36)(34,39)(38,40);(1,3)(2,4)(5,8)(6,7)(11,20)(12,18)(14,22)(16,23)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,4)(2,3)(5,6)(7,8)(9,12,15,18)(10,14,19,22)(11,21,20,13)(16,17,23,24);(9,13,15,21)(10,17,19,24)(11,12,20,18)(14,16,22,23)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(9,14)(10,18)(11,17)(12,19)(13,16)(15,22)(20,24)(21,23)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839633);