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 | #57142 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(26,27)(31,33)(32,34)(39,40);(3,4)(6,8)(11,17)(15,21)(18,23)(22,24)(25,26,30,27)(28,31,36,33)(29,32,38,34)(35,39,37,40);(1,3,5,4)(2,6,7,8)(10,14)(15,21)(16,19)(22,24);(3,4)(6,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27,30,26)(28,31,36,33)(29,34,38,32)(35,39,37,40);(1,4,5,3)(2,6,7,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,28,30,36)(26,31,27,33)(29,37,38,35)(32,40,34,39);(1,2)(3,6)(4,8)(5,7)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,30,38)(26,32,27,34)(28,35,36,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=i,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10065599);