Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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×D4×D8 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(5,6)(7,10)(8,9)(11,12);(1,2,4,3)(6,9)(10,12)(13,14,17,19)(15,16,18,20);(1,3,4,2)(5,7,8,11)(6,10,9,12)(14,18)(15,19)(16,20);(13,15)(14,16)(17,18)(19,20);(14,15)(16,20)(18,19)Polycyclic presentation:
2^9|b^2=gi,c^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280182, #72283159, #72283295, … |
| maximal normal subgroup of | #72280182, #72283159, #72283295, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6486453);