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 |
| prime radical | #42433 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4319 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,7,6)(3,5,8,4)(10,14)(12,20)(15,17)(18,24)(25,26,30,34)(27,31,36,37)(28,32,38,40)(29,33,39,35);(1,3)(2,4)(5,6)(7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,30,36)(26,31,34,37)(28,33,38,35)(29,32,39,40);(9,10,13,14)(11,15,19,17)(12,16,20,21)(18,22,24,23)(25,27)(26,31)(28,35)(29,32)(30,36)(33,38)(34,37)(39,40);(1,4,7,5)(2,8,6,3)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24);(1,5,7,4)(2,8,6,3)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28)(26,32)(27,35)(29,31)(30,38)(33,36)(34,40)(37,39);(1,6,7,2)(3,5,8,4)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,33)(27,32)(28,37)(30,39)(31,38)(34,35)(36,40)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9701500);