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) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34964 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2039 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)(13,14,19,24,18,23,17,22)(15,25,27,21,26,20,16,28);(6,9)(7,11)(13,15,19,27,18,26,17,16)(14,20,24,28,23,25,22,21);(1,2,3,4)(13,16,17,26,18,27,19,15)(14,21,22,25,23,28,24,20);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,22,23,24)(15,16,26,27)(20,21,25,28);(6,9)(10,12)(14,23)(20,25)(21,28)(22,24)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6303519);