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 | #52164 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4399 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,26)(27,35)(28,38)(29,32)(30,36)(31,33)(34,39)(37,40);(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,27,31,29)(26,32,33,35)(28,36,39,37)(30,38,40,34);(1,3,5,8)(2,6,4,7)(12,20)(15,22)(18,24)(21,23)(26,33)(30,40)(32,35)(34,38);(1,4,5,2)(3,7,8,6)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,28,31,39)(26,34,33,38)(27,37,29,36)(30,32,40,35);(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,29,31,27)(26,35,33,32)(28,37,39,36)(30,34,40,38);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,30,31,40)(26,36,33,37)(27,38,29,34)(28,32,39,35)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=i,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8247030);