Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419895 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(7,8)(9,10)(11,21)(12,18,28,33)(13,16)(14,19)(15,20)(17,24)(22,32)(23,37,39,35)(25,31)(26,34,27,29)(30,38,40,36);(1,2,5,6)(3,7,4,8)(10,16)(11,22)(12,26)(13,14)(18,33)(19,20)(21,31)(24,25)(27,28)(30,35)(36,38)(37,40);(1,3,5,4)(2,7,6,8)(9,11,14,24,15,25,13,22)(10,17,16,21,20,32,19,31)(12,23,26,38,28,39,27,36)(18,30,29,35,33,40,34,37);(1,4,5,3)(2,7,6,8)(9,12)(10,18)(11,23)(13,27)(14,26)(15,28)(16,29)(17,30)(19,34)(20,33)(21,35)(22,36)(24,38)(25,39)(31,37)(32,40)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,c^2=efgi,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 353627);