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 | |
4380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,15,12)(11,16,14,18)(13,20,22,17)(19,24,21,23)(25,26,31,35)(27,36,37,32)(28,29,33,39)(30,34,38,40);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,18)(13,19)(14,15)(17,23)(20,24)(21,22)(25,27)(26,32)(28,38)(29,34)(30,33)(31,37)(35,36)(39,40);(1,3,5,4)(2,7,6,8)(10,12)(11,14)(17,20)(19,21)(27,37)(30,38)(32,36)(34,40);(9,12,15,10)(11,18,14,16)(13,17,22,20)(19,23,21,24)(25,28)(26,29)(27,38)(30,37)(31,33)(32,34)(35,39)(36,40);(1,4,5,3)(2,8,6,7)(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,29)(26,33)(27,34)(28,35)(30,32)(31,39)(36,38)(37,40);(9,14,15,11)(10,18,12,16)(13,21,22,19)(17,24,20,23)(25,30,31,38)(26,34,35,40)(27,28,37,33)(29,32,39,36)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7756007);