Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| 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 | |
632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,13,17)(11,15)(12,22)(14,18,26,29)(16,28)(19,36,23,39)(20,32)(21,31)(24,33)(25,35)(27,34)(30,37,40,38);(2,5)(6,7)(9,11,14,27,13,20,26,21)(10,15,29,31,17,32,18,34)(12,23,24,37,25,19,28,38)(16,30,35,36,33,40,22,39);(1,3,4,8)(2,6,5,7)(14,26)(16,33)(18,29)(19,23)(21,27)(24,28)(31,34)(36,39);(3,8)(6,7)(9,12)(10,16)(11,19)(13,25)(14,28)(15,30)(17,33)(18,35)(20,23)(21,38)(22,29)(24,26)(27,37)(31,39)(32,40)(34,36)Polycyclic presentation:
2^9|a^2=fg,b^2=efh,c^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=def,e^b=ei,g^b=gi,h^b=hi,d^c=de,h^c=hi,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387011);