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 | |
4434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,17)(11,12,19,15)(13,16,21,23)(18,20,24,22)(25,26,31,30)(27,28,35,32)(29,37,38,33)(34,39,40,36);(1,3,5,8)(2,6,4,7)(25,27,31,35)(26,32,30,28)(29,34,38,40)(33,39,37,36);(9,11,14,19)(10,15,17,12)(13,18,21,24)(16,22,23,20)(25,28,31,32)(26,27,30,35)(29,36,38,39)(33,34,37,40);(1,2,5,4)(3,6,8,7)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,28)(26,27)(29,36)(30,35)(31,32)(33,34)(37,40)(38,39);(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,22)(23,24)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,16,17,23)(11,18,19,24)(12,20,15,22)(25,30,31,26)(27,28,35,32)(29,33,38,37)(34,39,40,36)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=h,d^2=gi,f^2=ghi,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, 7808545);