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 | #45457 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,20)(12,18)(13,14)(15,17)(16,21)(19,23)(22,24)(25,26)(27,31)(28,36)(29,34)(30,32)(33,38)(35,40)(37,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,16)(12,13)(14,24)(15,21)(17,20)(18,19)(22,23)(25,27,30,29)(26,31,32,34)(28,35,38,37)(33,39,36,40);(10,17)(11,21)(12,22)(14,19)(26,32)(31,34)(33,36)(39,40);(1,3,4,7)(2,5,6,8)(9,12,15,22)(10,18,17,24)(11,13,21,23)(14,16,19,20)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,38)(37,40);(9,13)(10,19)(11,12)(14,17)(15,23)(16,18)(20,24)(21,22)(25,28)(26,33)(27,35)(29,37)(30,38)(31,39)(32,36)(34,40);(9,14,15,19)(10,13,17,23)(11,18,21,24)(12,16,22,20)(25,29,30,27)(26,34,32,31)(28,37,38,35)(33,40,36,39)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,d^2=g,f^2=h,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, 7731594);