Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×C4⋊C8 |
| prime radical | C24 | C4⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,9)(7,11)(10,12)(13,14,15,18,16,19,21,24,17,20,22,25,23,26,27,28);(5,6,7,9,8,10,11,12);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=d,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg176 conjugacy classes — view the full character table.
| maximal subgroup of | #72286585, #72286592, #72287193, … |
| maximal normal subgroup of | #72286585, #72286592, #72287193, … |
| maximal quotient of | #72286592, #72286618 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41848);