Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,15,17,20,25,22)(16,19,24,21,23,26,28,27);(6,9)(10,12)(13,15,18,20,17,22,25,14)(16,21,24,26,23,27,28,19);(1,2,3,4);(5,7,8,11)(6,10,9,12)(13,16,17,23)(14,19,20,26)(15,21,22,27)(18,24,25,28);(16,23)(19,26)(21,27)(24,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,e^d=ei140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279357);