Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#468 |
| 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 | #36124 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,20)(17,22)(18,23)(19,24)(25,28)(26,27);(1,2)(3,4)(6,9)(10,12)(13,15,18,16,19,26,17,25)(14,20,22,21,24,28,23,27);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,26,19,25,17,15)(14,21,22,28,24,27,23,20);(1,2)(3,4)(13,17,19,18)(14,22,24,23)(15,25,26,16)(20,21,28,27);(2,4)Polycyclic presentation:
2^9|b^2=gi,c^2=f,d^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=ehi,d^c=dhi,e^d=ehi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72257549, #72257722, #72263865 |
| maximal normal subgroup of | #72257549, #72257722, #72263865 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6486601);