Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470225 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #62773 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4099 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,8)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23);(1,3)(2,5)(4,7)(6,8)(11,17)(14,21)(18,23)(22,24);(11,17)(14,21)(18,23)(22,24);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)Polycyclic presentation:
2^9|a^2=h,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,g^b=gi,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72238110, #72238184, #72256552, … |
| maximal normal subgroup of | #72238110, #72238184, #72256552, … |
| maximal quotient of | #72257617, #72316517, #72317689, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6403459);