Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417682 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
971 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(6,7)(10,16)(11,20)(12,24)(13,14)(19,28)(21,36)(22,23)(25,30)(26,27)(29,32)(31,37)(39,40);(1,2,4,5)(3,7,8,6)(9,10,15,19)(11,17,23,33)(12,25,27,30)(13,28,14,16)(18,24,34,26)(20,29,22,32)(21,37,36,39)(31,35,40,38);(9,11,15,23)(10,17,19,33)(12,21,27,36)(13,22,14,20)(16,29,28,32)(18,31,34,40)(24,38,26,35)(25,39,30,37);(1,3,4,8)(2,6,5,7)(9,12,15,27)(10,18,19,34)(11,21,23,36)(13,24,14,26)(16,30,28,25)(17,31,33,40)(20,35,22,38)(29,37,32,39)Polycyclic presentation:
2^9|b^2=ef,c^2=i,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336843);