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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4443 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)(9,10,15,14)(11,18,12,16)(13,17,21,22)(19,24,20,23)(25,26)(27,33)(28,32)(29,38)(30,35)(31,36)(34,39)(37,40);(1,3,5,8)(2,6,7,4)(11,12)(13,21)(16,18)(17,22)(25,27,31,28)(26,32,36,33)(29,37,39,35)(30,34,40,38);(1,3)(2,4)(5,8)(6,7)(9,11,15,12)(10,16,14,18)(13,19,21,20)(17,23,22,24)(25,27,31,28)(26,33,36,32)(29,35,39,37)(30,34,40,38);(1,4)(2,3)(5,6)(7,8)(9,12,15,11)(10,16,14,18)(13,20,21,19)(17,23,22,24)(25,28,31,27)(26,33,36,32)(29,37,39,35)(30,34,40,38);(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,29,31,39)(26,34,36,38)(27,35,28,37)(30,33,40,32);(9,14,15,10)(11,18,12,16)(13,22,21,17)(19,24,20,23)(25,30,31,40)(26,35,36,37)(27,34,28,38)(29,33,39,32)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gi,d^2=gi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8416085);