Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421163 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,19,15,20,14,17)(11,21,22,16,25,33,24,31)(12,26,29,18,30,35,28,36)(23,39,38,37,40,32,27,34);(2,5)(3,7)(9,11,14,24,15,25,13,22)(10,16,19,33,20,31,17,21)(12,27,29,23,30,38,28,40)(18,34,26,32,36,37,35,39);(10,17)(11,22)(13,14)(16,31)(18,26)(19,20)(23,38)(24,25)(27,40)(28,29)(32,39)(35,36);(1,3,4,7)(2,6,5,8)(9,12,15,30)(10,18)(11,23,25,40)(13,29,14,28)(16,32)(17,26)(19,35)(20,36)(21,37)(22,38,24,27)(31,39)(33,34)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 360177);