Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2329 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)(5,6,8,9)(7,10,11,12)(13,14,20,27,19,23,18,16)(15,24,28,21,25,26,17,22);(6,9)(10,12)(13,15,19,25)(14,21,23,22)(16,26,27,24)(17,18,28,20);(1,2,3,4)(13,16,18,23,19,27,20,14)(15,22,17,26,25,21,28,24);(5,7,8,11)(6,10,9,12)(13,17,18,25,19,28,20,15)(14,22,16,26,23,21,27,24);(14,23)(16,27)(21,22)(24,26)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=gh,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367210);