Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454664 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
947 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,14,19)(11,20)(12,25,29,36)(13,17,30,18)(15,23)(16,27,35,28)(21,31)(22,33)(24,32)(26,39)(34,37)(38,40);(1,3,4,7)(2,5,6,8)(9,11,14,24)(10,15,19,33)(12,26,29,37)(13,23,30,22)(16,34,35,39)(17,32,18,20)(21,27,38,28)(25,31,36,40);(3,7)(5,8)(9,12)(10,16)(11,21)(13,27)(14,29)(15,31)(17,25)(18,36)(19,35)(20,34)(22,37)(23,26)(24,38)(28,30)(32,39)(33,40);(10,17)(12,27)(13,30)(15,32)(16,35)(18,19)(20,33)(21,37)(22,23)(26,38)(28,29)(34,39)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381042);