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×C4⋊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 | |
1442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,12,17,13,16,20,23,14,18,21,24,11,15,19,22);(2,4)(10,15,18,16)(17,22,24,23);(6,8)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,f^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 | #72287595, #72287665, #72287668 |
| maximal normal subgroup of | #72287595, #72287665, #72287668 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53564);