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×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
893 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)(11,21,25,32)(12,26)(13,17)(14,19)(15,20)(16,22,30,24)(18,28)(23,31,38,39)(27,34)(29,33)(35,37,40,36);(1,3,4,7)(2,5,6,8)(9,11,15,25)(10,16,20,30)(12,23,29,38)(13,24,14,22)(17,32,19,21)(18,31,34,39)(26,35,33,40)(27,36,28,37);(3,7)(5,8)(10,17)(11,22)(12,27)(13,14)(16,30)(19,20)(24,25)(26,33)(28,29)(31,35)(36,37)(39,40);(9,12,15,29)(10,18,20,34)(11,23,25,38)(13,28,14,27)(16,31,30,39)(17,33,19,26)(21,35,32,40)(22,36,24,37)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,d^2=i,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, 381044);