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 | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
729 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(7,8)(9,10)(11,18)(12,26)(13,28,31,37)(14,21)(15,20)(16,22)(17,24)(19,30,36,29)(23,38,39,32)(25,33)(27,35,40,34);(1,2,5,6)(3,7,4,8)(9,11,16,25)(10,17,22,26)(12,15,24,14)(13,27,31,40)(18,20,33,21)(19,35,36,34)(23,30,39,29)(28,32,37,38);(1,3,5,4)(2,7,6,8)(9,12,15,25,16,24,14,11)(10,18,21,17,22,33,20,26)(13,27,29,39,31,40,30,23)(19,34,28,32,36,35,37,38);(1,4,5,3)(2,7,6,8)(9,13,16,31)(10,19,22,36)(11,23,25,39)(12,27,24,40)(14,30,15,29)(17,32,26,38)(18,34,33,35)(20,37,21,28)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,c^2=efgi,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, 353625);