Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C32 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4×C16 | C22 |
| commutator | C2 | C8×C32 |
| socle, prime radical | C23 | C4×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 44 points:
(1,2)(3,4)(13,14,15,19,16,20,23,29,17,21,24,30,26,32,35,39,18,22,25,31,27,33,36,40,28,34,37,41,38,42,43,44);(2,4)(5,6,7,9,8,10,11,12)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,h^2=i,b^a=bc320 conjugacy classes — view the full character table.
| maximal subgroup of | #72234287 |
| maximal normal subgroup of | #72234287 |
| maximal quotient of | #72234282, #72234299, #72234310 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1185);