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 | #40521 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4560 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)(9,10,14,16)(11,17,18,12)(13,15,23,24)(19,22,21,20)(25,26)(27,36)(28,30)(29,40)(31,33)(32,37)(34,39)(35,38);(1,2)(3,7)(4,8)(5,6)(9,11)(10,12)(13,21)(14,18)(15,22)(16,17)(19,23)(20,24)(25,27,31,37)(26,32,33,36)(28,35,39,40)(29,34,38,30);(1,3,5,4)(2,7,6,8)(11,18)(12,17)(19,21)(20,22)(26,33)(27,37)(28,39)(29,38);(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24)(25,28)(26,34)(27,35)(29,36)(30,33)(31,39)(32,38)(37,40);(9,12)(10,11)(13,22)(14,17)(15,21)(16,18)(19,24)(20,23)(25,29)(26,35)(27,34)(28,36)(30,37)(31,38)(32,39)(33,40);(1,4,5,3)(2,8,6,7)(9,13,14,23)(10,15,16,24)(11,19,18,21)(12,20,17,22)(25,30,31,34)(26,28,33,39)(27,38,37,29)(32,40,36,35)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7881156);