Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C2×#392 | C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,8)(13,14,16,15,17,18,20,19);(2,4)(9,10)(11,12);(6,8)(10,12);(13,15,16,18,17,19,20,14)Polycyclic presentation:
2^9|a^2=h,d^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg128 conjugacy classes — view the full character table.
| maximal subgroup of | #72264216, #72264237, #72264277, … |
| maximal normal subgroup of | #72264216, #72264237, #72264277, … |
| maximal quotient of | #72321139, #72321162, #72322715 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266191);