Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C25 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10,13,19,15,18,29,34,16,20,14,30,12,27,28,39)(11,21,23,35,25,33,37,40,26,36,24,38,22,17,31,32);(2,4)(9,11,14,24,15,25,28,31,16,26,13,23,12,22,29,37)(10,17,19,32,18,21,34,35,20,33,30,40,27,36,39,38);(6,8)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi176 conjugacy classes — view the full character table.
| maximal subgroup of | #72286958, #72287066, #72287665, … |
| maximal normal subgroup of | #72286958, #72287066, #72287665, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53589);