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 | |
4506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,26,31,33)(27,30,36,32)(28,39,38,34)(29,40,37,35);(1,2)(3,7)(4,8)(5,6)(25,27)(26,32)(28,37)(29,38)(30,33)(31,36)(34,35)(39,40);(1,3,5,4)(2,7,6,8)(10,15)(11,16)(17,22)(18,23)(26,33)(27,36)(28,38)(35,40);(1,4,5,3)(2,8,6,7)(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,28,31,38)(26,34,33,39)(27,37,36,29)(30,40,32,35);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,29,31,37)(26,35,33,40)(27,38,36,28)(30,39,32,34);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,30)(26,36)(27,33)(28,40)(29,34)(31,32)(35,38)(37,39)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=ghi,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7824429);