Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#447 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×D42 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,12,15)(11,14,16,13)(17,18)(19,21)(20,22)(23,24);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,16)(14,15)(18,21)(22,23);(11,16)(13,14)(18,22)(20,24)(21,23);(9,11)(10,14)(12,16)(13,15);(11,16)(13,14)Polycyclic presentation:
2^9|a^2=fi,b^2=fhi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=eh95 conjugacy classes — view the full character table.
| maximal subgroup of | #72264149, #72282345, #72283585, … |
| maximal normal subgroup of | #72264149, #72282345, #72283585, … |
| maximal quotient of | #72264170, D4×#1864, #72305704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6504445);