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 | #43927 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4141 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,5)(7,8)(9,10,15,18)(11,14,17,12)(13,16,22,24)(19,23,21,20)(25,26,30,34)(27,31,36,37)(28,32,38,40)(29,33,39,35);(1,2,6,4)(3,8,5,7)(9,11,15,17)(10,12,18,14)(13,21,22,19)(16,23,24,20)(25,27,30,36)(26,31,34,37)(28,33,38,35)(29,32,39,40);(1,3,6,5)(2,7,4,8)(9,12,15,14)(10,11,18,17)(13,20,22,23)(16,19,24,21)(25,27)(26,31)(28,35)(29,32)(30,36)(33,38)(34,37)(39,40);(1,2,6,4)(3,8,5,7)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24);(1,4,6,2)(3,8,5,7)(9,13)(10,16)(11,19)(12,20)(14,23)(15,22)(17,21)(18,24)(25,28)(26,32)(27,35)(29,31)(30,38)(33,36)(34,40)(37,39);(1,5,6,3)(2,7,4,8)(9,14,15,12)(10,17,18,11)(13,20,22,23)(16,19,24,21)(25,29)(26,33)(27,32)(28,37)(30,39)(31,38)(34,35)(36,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=h,e^2=h,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, 9701421);