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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4241 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(13,21)(16,20)(18,24)(22,23)(25,26)(27,33)(28,36)(29,31)(30,34)(32,37)(35,38)(39,40);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,22)(20,21)(23,24)(25,27,29,34)(26,30,31,33)(28,35,37,40)(32,38,36,39);(10,12)(15,17)(16,20)(22,23)(26,31)(27,34)(32,36)(35,40);(1,2,5,4)(3,6,8,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,26)(27,30)(28,32)(29,31)(33,34)(35,38)(36,37)(39,40);(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,39)(37,40);(1,4,5,2)(3,7,8,6)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,28,29,37)(26,32,31,36)(27,35,34,40)(30,38,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=gi,e^2=hi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7626485);