Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394358 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60425 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1325 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,20)(12,16)(13,18)(14,17)(15,22)(19,27)(21,28)(23,30)(24,34)(25,32)(26,31)(29,36)(33,38)(35,39)(37,40);(1,3,5,4)(2,6,8,7)(9,11,13,22)(10,15,18,20)(12,24,25,21)(14,23,27,36)(16,28,32,34)(17,29,19,30)(26,37,38,35)(31,39,33,40);(1,3,5,4)(2,7,8,6)(9,12,13,25)(10,16,18,32)(11,21)(14,26,27,38)(15,28)(17,31,19,33)(20,34)(22,24)(23,35)(29,39)(30,40)(36,37);(1,4,5,3)(2,6,8,7)(10,17)(14,27)(15,29)(16,31)(18,19)(20,30)(23,36)(26,38)(28,39)(32,33)(34,40)(35,37)Polycyclic presentation:
2^9|b^2=e,c^2=ef,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305749);