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 | #52973 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4327 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10)(11,14)(12,19)(13,15)(16,20)(17,23)(18,21)(22,24)(25,26,30,34)(27,32,28,31)(29,33,37,40)(35,39,36,38);(1,2,5,3)(4,6,8,7)(9,11,13,18)(10,14,15,21)(12,17,20,24)(16,22,19,23)(25,27,30,28)(26,31,34,32)(29,35,37,36)(33,38,40,39);(2,3)(6,7)(10,15)(14,21)(16,19)(22,23)(29,37)(33,40)(35,36)(38,39);(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,23)(20,24)(25,28,30,27)(26,32,34,31)(29,36,37,35)(33,39,40,38);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,16)(13,15)(17,22)(18,21)(19,20)(23,24)(25,29,30,37)(26,33,34,40)(27,35,28,36)(31,38,32,39);(1,4)(2,6)(3,7)(5,8)(9,12,13,20)(10,16,15,19)(11,17,18,24)(14,22,21,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,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, 8002147);