Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500416 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1954 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)(15,17)(16,23)(18,21)(19,22)(20,26)(24,25)(27,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,22,19,21,17,14)(16,24,26,20,25,23,27,28);(2,4)(6,9)(7,11)(13,16,17,27,19,25,18,26)(14,20,21,24,22,28,15,23);(5,7,8,11)(6,10,9,12)(16,25)(20,28)(23,24)(26,27);(6,9)(10,12)(13,17,19,18)(14,15,22,21)(16,26,25,27)(20,24,28,23)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6421615);