Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419919 |
| 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 | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(7,8)(9,10,16,20)(11,21,26,33)(12,15,19,14)(13,27)(17,25,22,24)(18,29)(23,38)(28,34)(30,36)(31,35)(32,40)(37,39);(1,2,5,6)(3,7,4,8)(9,11)(10,17)(12,21)(13,28)(14,25)(15,24)(16,26)(18,34)(19,33)(20,22)(23,29)(27,32)(30,40)(31,39)(35,38)(36,37);(1,3,5,4)(2,7,6,8)(9,12,15,20,16,19,14,10)(11,22,25,21,26,17,24,33)(13,27,29,36,31,35,30,18)(23,38,28,37,40,32,39,34);(1,4,5,3)(2,7,6,8)(9,13)(10,18)(11,23)(12,27)(14,30)(15,29)(16,31)(17,32)(19,35)(20,36)(21,37)(22,38)(24,39)(25,28)(26,40)(33,34)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,c^2=efg,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,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 353624);