Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7221 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #340 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | C2×#340 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,21,24,34)(12,16)(13,28)(14,20)(15,23,31,25)(17,26)(18,35)(19,27)(22,36,39,37)(29,33)(30,38,32,40);(1,3)(2,5)(4,8)(6,7)(9,11,13,25,12,22,26,40,14,24,29,23,27,39,35,38)(10,15,18,21,19,32,33,37,20,31,17,34,16,30,28,36);(9,12,14,27)(10,16,20,19)(11,22,24,39)(13,26,29,35)(15,30,31,32)(17,33,18,28)(21,36,34,37)(23,38,25,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=ghi,f^2=ghi,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,d^b=di,e^b=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34544);