Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C2×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C8×D4 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
605 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,18,13,19,28,36,15,20,30,37,29,17,33,35)(11,12,22,25,21,26,32,39,23,27,34,40,38,24,16,31);(2,5)(6,8)(9,11,14,22,13,21,28,32,15,23,30,34,29,38,33,16)(10,12,18,25,19,26,36,39,20,27,37,40,17,24,35,31);(9,12)(10,16)(11,18)(13,26)(14,25)(15,27)(17,34)(19,22)(20,32)(21,36)(23,37)(24,29)(28,39)(30,40)(31,33)(35,38);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,f^c=fi152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402145);