Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| 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 | |
795 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)(9,10,15,19,13,18,14,27)(11,20,23,31,22,16,28,35)(12,17,29,34,25,33,26,36)(21,30,24,40,38,37,39,32);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16,18,20)(12,24)(13,22)(14,28)(15,23)(17,32,33,40)(19,35,27,31)(21,26)(25,39)(29,38)(30,36,37,34);(2,6)(5,8)(9,12,13,25)(10,17,18,33)(11,21,22,38)(14,29,15,26)(16,30,20,37)(19,36,27,34)(23,39,28,24)(31,40,35,32);(3,7)(5,8)(10,18)(11,22)(14,15)(16,20)(17,33)(24,39)(26,29)(32,40)Polycyclic presentation:
2^9|a^2=fg,b^2=eh,c^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,g^c=gi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386936);