Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2029 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(13,14,19,16)(15,24,27,28)(17,22,18,23)(20,26,21,25);(2,4)(5,6)(7,11)(8,9)(10,12)(13,15,18,26,19,27,17,25)(14,20,22,28,16,21,23,24);(6,9)(10,11)(13,15,17,25,19,27,18,26)(14,21,23,28,16,20,22,24);(5,7,8,12)(6,10,9,11)(13,16,19,14)(15,20,27,21)(17,22,18,23)(24,26,28,25);(13,17,19,18)(14,22,16,23)(15,25,27,26)(20,28,21,24)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gi,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370990);