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 | #40006 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(26,32)(27,30)(29,37)(36,40);(1,2,5,6)(3,7,4,8)(25,26)(27,28)(29,34)(30,33)(31,32)(35,36)(37,39)(38,40);(1,3,5,4)(2,7,6,8)(10,15)(11,16)(17,22)(18,23)(25,27,31,30)(26,33,32,28)(29,38,37,35)(34,36,39,40);(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,28,31,33)(26,27,32,30)(29,36,37,40)(34,35,39,38);(1,4,5,3)(2,8,6,7)(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,29,31,37)(26,34,32,39)(27,35,30,38)(28,36,33,40);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,30,31,27)(26,33,32,28)(29,38,37,35)(34,40,39,36)Polycyclic presentation:
2^9|b^2=gi,c^2=g,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7616855);