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 | #41052 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4255 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)(3,4)(5,8)(6,7)(9,10,15,12)(11,19,20,16)(13,17,24,21)(14,18,23,22)(25,26,30,33)(27,31,35,28)(29,37,39,32)(34,36,40,38);(1,3)(2,4)(5,6)(7,8)(9,11,15,20)(10,16,12,19)(13,23,24,14)(17,18,21,22)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,39)(38,40);(1,2,5,8)(3,7,6,4)(25,27,30,35)(26,28,33,31)(29,34,39,40)(32,36,37,38);(1,4)(2,6)(3,8)(5,7)(9,12,15,10)(11,16,20,19)(13,21,24,17)(14,22,23,18)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,37)(39,40);(1,4)(2,3)(5,7)(6,8)(9,13,15,24)(10,17,12,21)(11,14,20,23)(16,22,19,18)(25,28,30,31)(26,27,33,35)(29,38,39,36)(32,40,37,34);(1,3)(2,7)(4,8)(5,6)(9,14)(10,18)(11,13)(12,22)(15,23)(16,21)(17,19)(20,24)(25,29)(26,32)(27,34)(28,36)(30,39)(31,38)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=h,d^2=ghi,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9854510);