Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418488 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| 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 | |
903 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,18,15,19,13,20)(11,21,24,16,26,23,25,33)(12,27,29,35,31,17,30,36)(22,37,28,32,39,34,38,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,28)(13,25)(14,24)(15,26)(17,34)(18,23)(19,33)(20,21)(22,30)(27,37)(29,39)(31,38)(32,35)(36,40);(1,4,5,2)(3,7,8,6)(9,12,14,30,15,31,13,29)(10,17,18,35,19,27,20,36)(11,22,25,28,26,39,24,38)(16,32,21,34,33,40,23,37);(3,8)(6,7)(9,10)(11,23)(12,17)(13,20)(14,18)(15,19)(16,25)(21,26)(22,37)(24,33)(27,31)(28,32)(29,36)(30,35)(34,39)(38,40)Polycyclic presentation:
2^9|a^2=egh,b^2=efg,c^2=efi,e^2=i,g^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,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, 359933);