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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4065 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)(9,10,15,17)(11,19,13,16)(12,18,22,23)(14,20,24,21)(25,26,30,34)(27,32,28,31)(29,37,38,33)(35,39,36,40);(1,3,5,8)(2,6,7,4)(9,11,15,13)(10,16,17,19)(12,21,22,20)(14,18,24,23)(25,26)(27,32)(28,31)(29,33)(30,34)(35,40)(36,39)(37,38);(1,3)(2,4)(5,8)(6,7)(10,17)(12,22)(16,19)(20,21)(25,27,30,28)(26,31,34,32)(29,35,38,36)(33,39,37,40);(1,4)(2,3)(5,6)(7,8)(9,12)(10,18)(11,20)(13,21)(14,16)(15,22)(17,23)(19,24)(25,28,30,27)(26,32,34,31)(29,36,38,35)(33,40,37,39);(9,13,15,11)(10,19,17,16)(12,21,22,20)(14,18,24,23)(25,29,30,38)(26,33,34,37)(27,35,28,36)(31,39,32,40);(9,14,15,24)(10,20,17,21)(11,18,13,23)(12,19,22,16)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=ghi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8658688);